Global Attractors in Partial Differential Equations
G. Raugel
CNRS et Université de Paris-Sud
Analyse Numérique et EDP, UMR 8628
Bâtiment 425
F-91405 Orsay Cedex, France
[email protected]
1. Introduction
This survey is devoted to an introduction to the theory of global attractors for semigroups defined on infinite dimensional spaces, which has mainly been developed in the
last three decades. A first purpose here is to describe the main ingredients leading to
the existence of a global attractor. Once a global attractor is obtained, the question
arises if it has special regularity properties, a particular shape etc... or if it has a finitedimensional character. The second objective is thus naturally to give some of the most
important properties of global attractors. Finally, we want to show the relevance of the
abstract theory in applications to evolutionary equations.
Clearly, we can here neither describe all the related questions and results, nor give
the detailed proofs of the main statements, although they are often very instructive.
To keep the text elementary and self-contained, we have recalled all the needed basic
concepts in the theory of dynamical systems and have included some proofs. In order to
illustrate the general abstract results, we have chosen to discuss few equations, but in
details, rather than to give a catalogue of applications to partial differential equations
and functional differential equations.
The dynamical systems that arise in physics, chemistry or biology, are often generated by a partial differential equation or a functional differential equation and thus the
underlying state space is infinite-dimensional. Usually these systems are either conservative or exhibit some dissipation. In the last case, one can hope to reduce the study of
the flow to a bounded (or even compact) attracting set or global attractor, that contains
much of the relevant information about the flow and often has some finite-dimensional
character.
-2It is difficult to trace the origin of the concepts of dissipation and attractor. The
word attractor, applied to a single invariant point, is ancient and probably appeared at
the beginning of the century. One can find it, for example, in the book of Coddington
and Levinson in 1955 [CoLe55] or in a paper of Mendelson in 1960 [Me60]. For a flow
on a locally compact metric space, attractors consisting of more than one point have
been studied by Auslander, Bhatia and Seibert in 1964 [ABS] (see the paper of [Mil] for
various definitions in the finite-dimensional case). Several different notions of attractors
are already found in the lecture notes of Bhatia and Hajek ([BhHa]) in 1969, for a
semi-flow on an infinite-dimensional space. In 1968, Gerstein and Krasnoselskii [GeKr]
studied the existence and properties of a maximal compact invariant set for the discrete
system generated by a compact map S on a Banach space. In 1971, Billotti and LaSalle
[BiLaS] described the maximal compact invariant set and proved stability results for
maps whose iterates were eventually compact. The specific notion of compact global
attractor, as used in this review, appeared in the papers of Oliva in the early 1970’s (see
[HMO]). The work of Ladyzenskaya ([La72], [La73]) in 1972 implied the existence of the
compact global attractor for the semi-flow generated by the two-dimensional NavierStokes equations. In the same year 1972, Hale, LaSalle and Slemrod [HaLaSSl] gave
general existence results of maximal compact invariant sets and introduced the concept
of asymptotically smooth systems.
Let us now describe more precisely the concepts of dissipation and global attractor.
In his study of the forced van der Pol equation, Levinson [Le44] introduced the concept
point dissipative for maps S on the space Rn . A map S is point dissipative if there
exists a bounded set B0 ⊂ Rn such that, for each x ∈ Rn , there exists an integer
n0 (x, B0 ) so that S n (x) ∈ B0 , for n ≥ n0 . Due to the local compactness of Rn , any
point dissipative map S is also bounded dissipative (or equivalently uniformly ultimately
bounded); that is, there exists a bounded set B0 ⊂ Rn such that, for each bounded
set B ⊂ Rn , there exists an integer n0 (B, B0) so that S n (B) ⊂ B0 , for n ≥ n0 . If
S is bounded dissipative, the local compactness of Rn also implies that the ω-limit
set ω(B) ≡ ∩m≥0 Cl(∪j≥m S j (B)) of any bounded set B is compact, invariant (i.e.
S(ω(B)) = ω(B)) and attracts B, that is, δRn (S n (B), ω(B)) → 0 as n → +∞, where
δRn (S n (B), ω(B)) = supx∈S n (B) inf y∈ω(B) kx−ykRn . Therefore, if S is point dissipative,
A = ω(B0 ) is the global attractor, that is, A is bounded, invariant and attracts every
bounded set B of Rn . Here, in addition, A is compact. Thus, in finite dimensions,
point dissipativness implies the existence of a compact global attractor. Note that this
-3definition of the global attractor implies that A is maximal with respect to inclusion
and hence is unique.
Unfortunately, when the underlying space X is not locally compact, there are examples where point dissipativness does not imply that the orbits of bounded sets are
bounded and where bounded dissipativness does not imply the existence of a global
attractor. So the following question arises: are there interesting classes of dynamical
sytems on non locally compact spaces that have properties similar to the ones mentioned
above for dynamical systems on Rn ? To have a theory comparable to the one for maps
on Rn , one must impose a type of smoothing property on the operator S : X → X. This
is done by assuming, for example, that S : X → X or an iterate of S is compact (see
[BiLaS]). More generally, it is sufficient to suppose that S is asymptotically smooth in
the terminology of Hale, Lasalle and Slemrod [HaLaSSl] or equivalently, asymptotically
compact in the terminology of Ladyzhenskaya [La87a].
In Section 2, we recall all the needed precise definitions, introduce the above concepts of dissipativness and asymptotic smooth or compact systems. We discuss some
implications between these notions. The fundamental theorem of existence of a compact
global attractor is stated and proved. Some basic properties like invariance, stability
and connectedness of compact global attractors are also discussed. Finally, a large part
of the section contains examples of asymptotically smooth systems.
Section 3 is devoted to a presentation of the most important properties of compact
global attractors. Compact global attractors are robust objects with respect to perturbations. We give several continuity properties of the global attractors with respect
to perturbation parameters and recall the stability of the flow on the global attractor
under perturbations for Morse-Smale systems. The mentioned properties play an important role in the study of systems depending on several physical parameters and also
in numerical approximations of these systems. Finally, we discuss the possibility of the
flow on the compact global attractor A being finite-dimensional by first showing that,
in most of the cases, A has finite Hausdorff or fractal dimension. The next question
of interest is the reduction of the study of the flow on A to the discussion of the flow
of some system on a finite-dimensional space. One effort in this direction is to assert
the existence of an inertial manifold, that is a finite-dimensional Lipschitzian positively
invariant manifold, that contains the global attractor. Unfortunately, the existence of
inertial manifolds is rare in the general class of systems arising in applications. Another approach is to show the existence of a finite number of “modes”, on which the
-4corresponding dynamics approximates the dynamics of the original system on A (for
example, Galerkin approximations). For evolutionary equations, this approach gives
regularity with respect to time of the flow on A and regularity in “the spatial variables”
when PDE’s are involved.
So far, one has not yet given a description of the flow on the global attractor.
In the general case, a qualitative description of the global attractor seems difficult.
Section 4 is devoted to the class of gradient systems, that is systems which admit a
strict Lyapunov functional. In this case, due to the invariance principle of LaSalle, the
global attractor A, if it exists, is the unstable set of the set of equilibrium points. If
the equilibrium points are all hyperbolic, then A is the union of the unstable manifolds
of each equilibrium point. Applications of the general abstract theory, in the frame of
gradient systems, are then given to FDE’s and to two representative classes of scalar
partial differential equations, the reaction-diffusion equations and the (weakly) damped
wave equations. Special emphasis is made on the scalar reaction-diffusion equation
defined on a bounded interval of R and provided with separated boundary conditions.
In this case, a result of Henry ([He85b], [An86]) says that the stable and unstable
manifolds are always tranversal, which means that the global dynamical behaviour can
only change by bifurcations of the equilibria. This important property was the starting
point for the precise qualitative description of the flow on the global attractor. In
this one-dimensional case, special properties like the strong maximum principle, the
Sturm-Liouville theory and the Jordan curve theorem play a primordial role.
Finally, in Section 5, we illustrate the abstract theory of global attractors given in
Section 2, by studying weakly damped dispersive equations, the prototype of which is
the weakly damped Schrödinger equation.
Many topics have been left on the side, including the non autonomous evolutionary
equations leading to the notions of processes and skew-product semi-flows (see [Da75],
[Sell71], [MiSe], [Har91], [Vi92], [ChVi1], etc . . .), the generalization of the concept
of attractor to multivalued mappings (see [Ba2] for instance), the notion of random
attractors for dissipative stochastic dynamical systems (see [CFl], [Deb] for example).
Only few applications to the class of retarded functional differential equations have been
given below (see [HVL] and [Nu00]). Finally, for further readings on global attractors
and more examples, the reader should consult the books [BV89b], [Hal88], [Te], [La91],
[ChVi2], [SeYou], for example.
-5Acknowledgements. I am indebted to Th. Gallay, O. Goubet and E. Titi for their
helpful suggestions. I especially express my gratitude to Jack Hale, who introduced
me into the field of dynamical systems and over the years did not spare his time to
discuss with me and answer my numerous questions. I also thank him for the advices
and comments on this manuscript. Finally my thanks go to Bernold Fiedler, who gave
me the opportunity to write this text.
2. Fundamental Concepts
In this section, (X, d) (or simply X) denotes a metric space, with distance d. We use
the semi-distance δX (·, ·) defined on the subsets of X by
δX (x, A) = inf d(x, a) ,
a∈A
∀x ∈ X ,
∀A ⊂ X ,
δX (A, B) = sup inf d(a, b) = sup δX (a, B) ,
a∈A b∈B
a∈A
∀A , B ⊂ X .
For any subset A of X and any positive number ε, we introduce the open neighbourhood
NX (A, ε) = {z ∈ X | δX (z, A) < ε} (resp. the closed neighbourhood N X (A, ε) = {z ∈
X | δX (z, A) ≤ ε}). Finally, we define the Hausdorff distance HdistX (A, B) , for any
subsets A, B of X by
HdistX (A, B) = max(δX (A, B), δX (B, A)) .
In what follows, we mainly concentrate on continuous dynamical systems, or continuous semigroups, S(t), t ≥ 0 on X, whose definition we now recall.
Definition 2.1. A continuous dynamical system or continuous semigroup on X is a
one-parameter family of mappings S(t), t ≥ 0 from X into X such that
1) S(0) = I ;
2) S(t + s) = S(t)S(s) for any t, s ≥ 0;
3) for any t ≥ 0, S(t) ∈ C 0 (X, X);
4) for any u ∈ X, t 7→ S(t)u ∈ C 0 ((0, +∞), X).
If the mappings S(t) from X into X are defined for t ∈ R, if the properties 2), 3) hold
for any t, s ∈ R and if, in 4), (0, +∞) can be replaced by R, then S(t), t ∈ R is a
continuous group. A one-parameter family of mappings S(t), t ≥ 0, satisfying only the
properties 1), 2) and 3) will be simply called ”a semigroup”.
-6We recall that, if S ∈ C 0 (X, X), the family S n , n ∈ N, is called a discrete dynamical
system or discrete semigroup. If S is a C 0 -diffeomorphism from X to X, then the family
S m , m ∈ Z, forms a discrete group. Most of the properties described below are also
valid for discrete dynamical systems. In the sequel, if we do not want to distinguish
between discrete dynamical systems and (non discrete) semigroups, we simply refer to
a semigroup S or S(t), t ∈ G+ , where G+ is either [0, +∞) or the set of nonnegative
integers N. Hereafter, G denotes either R or Z.
The first example of continuous semigroups is given by ordinary differential equations ẋ = f (x), x ∈ Rn , where f : x ∈ Rn 7→ f (x) ∈ Rn is a globally Lipschitzian
mapping. Another basic example is the class of retarded functional differential equations (see [HVL]). Evolutionary partial differential equations also give rise to continuous
semigroups as shown in the following model example.
Example 2.2. Let X, Y be two Banach spaces, such that Y ⊂ X, with continuous
injection. Let Σ0 (t) be a linear C 0 - semigroup in X with infinitesimal generator A and
f : Y → X be a Lipschitzian mapping on the bounded sets of Y . We assume that,
either,
1) Y = X,
or
2) Σ0 (t) is an analytic semigroup on X and Y = X α = D((λId − A)α ), where α ∈ [0, 1)
and λ is an appropriate real number.
We consider the semilinear differential equation in Y ,
du(t)
= Au(t) + f (u(t)) , t > 0 , u(0) = u0 ∈ Y .
(2.1)
dt
It is well-known that this equation has a unique mild solution u ∈ C 0 ([0, T ∗ (u0 )), Y ),
where T ∗ (u0 ) ∈ (0, +∞]. If T ∗ (u0 ) = +∞, for any u0 ∈ Y , then the family of mappings
S(t) defined by S(t)u0 = u(t) is a continuous semigroup on Y . In particular, the
mapping (u0 , t) 7→ S(t)u0 is continuous from [0, +∞) × Y into Y . We recall that u(t)
is a mild solution of (2.1) if, for t ≥ 0,
Z t
S(t)u0 = Σ0 (t)u0 +
Σ0 (t − s)f (S(s)u0) ds ≡ Σ0 (t)u0 + U (t)u0 .
(2.2)
0
Under the same hypotheses as above, assume now that there exists a subset Z0 of Y
such that T ∗ (u0 ) = +∞, for any u0 ∈ Z0 , and that there is a positive constant C0 such
that
kS(t)u0 kY ≤ C0 , ∀ u0 ∈ Z0 , ∀ t ≥ 0 .
-7S
S
Y
We thus introduce the set Z = u0 ∈Z0 t≥0 {S(t)u0 } , equipped with the distance d
induced by the norm of Y . Then, (Z, d) is a complete metric space, S(t)Z defines a
continuous semigroup on Z and kS(t)u0 kY ≤ C0 for any u0 ∈ Z, t ≥ 0.
Remark. In the definition of continuous dynamical systems, several authors require
also that, for any u ∈ X, the mapping t ∈ [0, +∞) 7→ S(t)u ∈ X is continuous at
t = 0. Actually, this hypothesis is unnecessary most of the time. In Section 4 below
(see Equation (4.69)), we study an example where S(t) is not continuous at t = 0.
A result of [CM] implies that, if S(t) is a continuous dynamical system in the sense
of Definition 2.1, then the mapping (t, u) ∈ (0, +∞) × X 7→ S(t)u ∈ X is continuous.
If, moreover, the space X is locally compact and if, for any u ∈ X, t ∈ [0, +∞) 7→
S(t)u ∈ X is continuous at t = 0, then, by a theorem of [Do], the mapping (t, u) ∈
[0, +∞) × X 7→ S(t)u ∈ X is continuous. If the space X is not locally compact, the
joint continuity of S(t)u at t = 0 may not be true (see the examples of [Ch] and [Ba2]).
2.1. Some definitions
In this subsection, we assume that S(t), t ∈ G+ , is a semigroup on X. Here we define
carefully the notions of invariance and attraction, which play a crucial role in the theory
of global attractors.
Definition 2.3. A set A is positively invariant if S(t)A ⊂ A, for any t ∈ G+ . The set
A is invariant if S(t)A = A, for any t ∈ G+ .
The following concept dealing with invariance and connectedness has been introduced in [LaS] and will be used later.
Definition. Let S be a semigroup of continuous maps from X into X. A closed
invariant subset A of X is said to be invariantly connected if it cannot be represented
as the union of two nonempty, disjoint, closed, positively invariant sets.
The positive orbit of x ∈ X is the set γ + (x) = {S(t)x | t ∈ G+ }. If E ⊂ X, the
positive orbit of E is the set
[
[
γ + (E) =
S(t)E =
γ + (z) .
t∈G+
z∈E
More generally, for τ ∈ G+ , we define the orbit after the time τ of E by
γτ+ (E) = γ + (S(τ )E) .
-8Let now I be an interval of R and S(t), t ≥ 0, a semigroup. We recall that a mapping
u from I into X is a trajectory (or orbit) of S(t) on I if u(t + s) = S(t)u(s), for any
s ∈ I and t ≥ 0 such that t + s ∈ I. In particular, if I = (−∞, 0] and u(0) = z ∈ X,
u is called a negative orbit through z and is often denoted by γ − (z) or uz . If I = R
and u(0) = z, then u is called a complete orbit through z and is often denoted by γ(z).
We let Γ− (z) be the set of all negative orbits through z. If Γ− (z) is not empty, it may
contain more than one negative orbit, because we have not assumed the property of
backward uniqueness. We also let Γ(z) = Γ− (z) ∪ γ + (z) be the set of all complete orbits
through z. In the same way, we define the sets γ − (E), Γ− (E) and Γ(E), for any subset
E of X. For later use, for any z ∈ X, we introduce the following set:
H(t, z) = {y ∈ X | there exists a negative orbit uz through z
such that uz (0) = z and uz (−t) = y} .
S
We remark that Γ− (z) = t≥0 H(t, z). Likewise, if E ⊂ X, we define the set H(t, E) =
S
S
−
z∈E H(t, z) and remark that Γ (E) =
t≥0 H(t, E).
In a similar way, replacing (−∞, 0] (resp. R) by (−∞, 0] ∩ Z (resp. Z), we define
the negative and complete orbits of maps S. In the framework of maps, it is very easy
to give examples of non backward uniqueness. Consider the non injective logistic map
S : [0, 1] → [0, 1], Sx = λx(1 − x), with 2 < λ ≤ 4. The point x0 = (λ − 1)/λ is a fixed
point of S and the point y = λ−1 satisfies Sy = x0 . The iterates S −n y ∈ (0, x0 ) are
well defined and γ(y) = {S −n y |, n = 0, 1, 2, . . .} ∪ {x0 } is a complete orbit trough x0 .
The proof of the following lemma is elementary.
Lemma 2.4. The set A ⊂ X is invariant for the semigroup S(t), t ∈ G+ if and only
if, for any a ∈ A, there exists a complete orbit ua through a, with ua (G+ ) ⊂ A. If the
semigroup S(t), t ≥ 0, is continuous, the complete orbits belong all to C 0 (R, A).
In general, there may exist an invariant set A, which does not contain all complete
orbits of S through each point in A. In the above example of the logistic map, the
invariant set A = {x0 } does not contain the complete orbit γ(y).
Proposition 2.5. Let S(t) be a continuous semigroup on X and A be a compact
invariant set. If the operators S(t) are injective on A, for t ≥ 0, then S(t)A is a
continuous group of continuous operators on A.
-9Proof. By Lemma 2.4, for any a ∈ A, there exists a complete orbit ua ∈ C 0 (R, A) such
that S(t)ua (−t) = a, for any t ≥ 0. Since S(t)A is one-to-one, we can set, for a ∈ A:
S(−t)a = S(t)−1 a = ua (−t) ,
Clearly, S(t)S(s) = S(t + s), for any t, s ∈ R. Moreover, for any t ≥ 0, S(t) : A → A is
a continuous bijection on the compact set A, and therefore is an isomorphism from A
to A.
Of primary importance in the theory of dynamical systems is the set
J = { bounded complete orbits of S} .
If this set J is bounded, then, by Lemma 2.4, it is the maximal bounded invariant set
that is; it is invariant, bounded and contains each bounded invariant set. If S has a
global attractor A, then A coincides with J . However, in the general case, J needs not
to be a global attractor, even if J is compact and attracts compact sets (see Example
2.24 below; examples involving continuous dynamical systems are also found in [Hal99]).
2.2. ω and α-limit sets
As indicated in the introduction, we are going to construct global attractors as ω-limit
sets of bounded sets. For this reason, we now recall the definition and main properties
of ω and α-limit sets.
Definition 2.6. Let E be a nonempty subset of X.
(i) We define the ω-limit set ω(E) of E as
ω(E) =
\
X
γ + (S(s)E)
=
\
s∈G+
s∈G+
(
X
[
S(t)E)
.
(2.3)
t≥s, t∈G+
(ii) We define the α-limit set α(E) of E as
α(E) =
\
s∈G+
(
[
X
H(t, E))
.
(2.4)
t≥s, t∈G+
Remark. Let z ∈ X be such that there exists a negative orbit uz through z. We define
the αuz -limit set αuz (z) of the orbit uz as
\
X
{uz (−t) | t ≥ s, t ∈ G+ } .
(2.5)
αuz (z) =
s∈G+
- 10 An equivalent description of the ω and α-limits sets is given in terms of limits of sequences as follows:
Lemma 2.7. (Characterization Lemma) Let E be a nonempty subset of X. Then,
ω(E) = {y ∈ X | there exist sequences tn ∈ G+ and zn ∈ E such that
tn
→
n→+∞
+∞ and S(tn )zn
α(E) = {y ∈ X | there exist sequences tn ∈
tn
→
n→+∞
+∞ , xn
→
n→+∞
→
y} ,
n→+∞
G+ , xn
∈ X and zn ∈ E such that
y where xn = uzn (−tn )
and uzn is a negative orbit through zn } .
Likewise, if z is a point in X such that there exists a negative orbit uz through z, then
αuz (z) = {y ∈ X | there exists a sequence tn ∈ G+ such that tn
and uz (−tn )
→
n→+∞
y} .
→
n→+∞
+∞
(2.6)
We remark that if E is a nonempty subset of X, we have the following inclusions,
for t ∈ G+ ,
ω(E) = ω(S(t)E) , α(E) ⊂ α(S(t)E) ,
(2.7)
S(t)ω(E) ⊂ ω(E) , S(t)α(E) ⊂ α(E)
S
Remark. If E is a nonempty subset of X, then, generally, ω(E) 6= z∈E ω(z). Indeed,
let us consider the flow S(t) generated by the following ordinary differential equation
ẏ = y(1 − y)(2 + y) .
For any y0 ∈ R, limt→+∞ S(t)y0 exists and limt→+∞ S(t)y0 = 1 if y0 > 0, S(t)0 = 0 and
limt→+∞ S(t)y0 = −2 if y0 < 0. Thus, ω(y0 ) = 1 if y0 > 0, ω(0) = 0 and ω(y0 ) = −2
if y0 < 0. However, for any t ≥ 0, S(t)[−2, 1] = [−2, 1] and therefore ω(E) = [−2, 1].
Example 2.8. The ω-limit set can be empty as the following example, which appeared
in the thesis of Cooperman [Coo] (see also [ChHa]) shows. Let H0 be the Banach space of
all real sequences x = {xi , i ≥ 1 | xi → 0 as i → +∞}, equipped with the norm kxkH0 =
supi≥1 |xi |. We introduce the map T : x = (x1 , x2 , . . .) ∈ H0 7→ (1, x1 , x2 , . . .) ∈ H0
and define the map U : H0 → H0 by U (x) = x/kxkH0 if kxkH0 > 1 and U (x) = x
- 11 if kxkH0 ≤ 1. Finally, we let S = T ◦ U . We remark that, since S n = T n ◦ U ,
for any x ∈ H0 , the first n terms in the sequence S n (x) are equal to 1. Clearly, for
any x0 ∈ H0 , the ω-limit set of x0 is empty. Indeed, by the characterisation lemma, if
ω(x0 ) 6= ∅, there exists y ∈ H0 and a sequence nj ∈ N, nj → +∞, such that S nj x0 → y.
Since y ∈ H0 , there exists i0 ∈ N such that, for i ≥ i0 , |yi | ≤ 1/2. But, for nj ≥ i0 ,
kS nj (x0 )−ykH0 ≥ 1/2, which contradicts the convergence of S nj x0 towards y. Likewise,
ω(E) = ∅ for any subset E of H0 .
Another obvious example is given by the flow generated on R by the ordinary differential
equation ẏ = 1.
Thus, we can wonder when the ω-limit sets are nonempty and which are their
properties.
Definition. Let A, E be two (nonempty) subsets of X. The set A is said to attract E
if
δX (S(t)E, A) → 0 ,
t→+∞
that is, for any ε > 0, there exists a time τ = τ (ε, A, E) ≥ 0 such that
S(t)E ⊂ N X (A, ε) ,
t ≥ τ .
The following properties are elementary, yet fundamental.
Lemma 2.9. Let E be a nonempty subset of X and S a semigroup on X. Assume that
ω(E) is nonempty, compact and attracts E, then the following properties hold:
1) ω(E) is invariant.
2) If moreover E is connected, ω(E) is invariantly connected. If, in addition, either
ω(E) ⊂ E or S(t) is a continuous semigroup, then ω(E) is connected.
Proof. Statement 1) as well as the connectedness of ω(E) in the case of a continuous
semigroup are well-known and their proofs can be found, for instance, in [Hal88, Chapters 2 and 3]. The connectedness of ω(E) in the case where ω(E) ⊂ E is shown in
[GoSa, Lemmas 4.1 and 4.2], for example. Here, adapting arguments given in [GoSa],
we prove Statement 2).
(i) We begin by showing by contradiction, that, if ω(E) is a nonempty compact set,
which attracts E, then ω(E) is invariantly connected, if E is connected. To simplify the notation, we assume, without loss of generality, that S is a discrete dynamical system. If ω(E) is not invariantly connected, then ω(E) = F1 ∪ F2 , where F1 ,
- 12 F2 are disjoint, nonempty, compact, positively invariant sets. We fix ε > 0 so that
NX (F1 , ε) ∩ NX (F2 , ε) = ∅. Since F1 and F2 are invariantly connected, the continuity
of the mapping S implies that there exists δ, 0 < δ < ε, such that S(N X (Fi , δ)) ⊂
NX (Fi , ε), for i = 1, 2. As ω(E) attracts E, there exists n0 ∈ N, such that,
S n E ⊂ N X (ω(E), δ) ,
∀n ≥ n0 .
(2.8)
In particular,
S n0 E ⊂ (N X (F1 , δ) ∩ S n0 E) ∪ (N X (F2 , δ) ∩ S n0 E) .
(2.9)
We note that, if x ∈ E and S n0 x ∈ N X (Fi , δ), then S n0 +1 x ∈ NX (Fi , ε) ∩ NX (ω(E), δ),
which implies that S n0 +1 x ∈ NX (Fi , δ). Thus, by recursion, S n x ∈ NX (Fi , δ), for
n ≥ n0 . It follows that, if there exists j, j = 1 or 2 such that N X (Fj , δ) ∩ S n0 E = ∅,
then N X (Fj , δ) ∩ S n E = ∅, for n ≥ n0 , which means that Fj = ∅. Hence Fi0 ≡
N X (Fi , δ)∩S n0 E is nonempty, for i = 1, 2. Thus, we have just proved that the connected
set S n0 E is the union of the two nonempty, closed, disjoint subsets F10 and F20 , which
is a contradiction. Therefore, ω(E) is invariantly connected.
(ii) To prove that ω(E) is connected when E is connected and ω(E) ⊂ E, we again
argue by contradiction. If ω(E) is not connected, then ω(E) = F1 ∪F2 , where F1 , F2 are
disjoint, nonempty, compact sets. We fix 0 < δ ≤ ε so that NX (F1 , ε) ∩ NX (F2 , ε) = ∅.
As in (i), there exists n0 ∈ N, such that the inclusions (2.8) and (2.9) hold. The property
ω(E) ⊂ E together with the invariance of ω(E) yields that ω(E) ⊂ S n0 E. Thus, we
deduce from (2.9) that Fi ⊂ N X (Fi , δ) ∩ S n0 E, for i = 1, 2. Hence, the connected set
S n0 E is the union of the two nonempty, closed, disjoint subsets F10 and F20 , which is a
contradiction. Therefore, ω(E) is connected.
(iii) Finally, we prove by contradiction that ω(E) is connected, when S(t) is a continuous
semigroup. Arguing as in (ii), we show that there exists n0 ∈ N, such that the inclusions
(2.8), for any t ≥ n0 , and (2.9) hold. To obtain a contradiction, we need to prove, as in
(ii), that Fi0 ≡ N X (Fi , δ) ∩ S n0 E is nonempty for i = 1, 2. Since S(t) is a continuous
semigroup and that F1 , F2 are compact sets, there exists a positive time τ such that
, for 0 ≤ t ≤ τ , S(t)(N X (Fi , δ)) ⊂ NX (Fi , ε), for i = 1, 2. Let x ∈ E be such that
S(n0 )x ∈ N X (Fi , δ). Then, for 0 ≤ t ≤ τ , S(n0 + t)x ∈ N X (ω(E), δ) ∩ NX (Fi , ε),
which implies that, for 0 ≤ t ≤ τ , S(n0 + t)x ∈ N X (Fi , δ). Thus, by recursion,
S(t)x ∈ NX (Fi , δ), for t ≥ n0 . We now conclude like in (i) that Fi0 ≡ N X (Fi , δ) ∩ S n0 E
is nonempty, for i = 1, 2.
- 13 Remarks 2.10.
(i) We deduce from the above lemma that, if E is connected and if ω(E) contains only
fixed points of S, then ω(E) is connected, which had been proved in [HR92b, Lemma
2.7]. This result is useful when one wants to show, in the case of gradient systems, that
the ω-limit set ω(x0 ) of an element x0 ∈ X is a single equilibrium point (see [HR92b]
and [BrP97b] as well as Section 4.1 below).
(ii) Adapting the proof of Lemma 2.9 one shows that, if the α-limit set α(E) of the
nonempty subset E of X is nonempty, compact and δX (H(t, E), α(E)) → 0 as t → +∞
in G+ , then α(E) is invariant. If moreover, H(t, E) is connected for any t ∈ G+ ,
α(E) is invariantly connected. If, in addition, either α(E) ⊂ E or S(t) is a continuous
semigroup, then α(E) is connected.
Of course, similar properties hold for the αuz -limit set of negative orbits uz through
z ∈ X.
The following property had already been proved for instance by Hale in 1969
([Hal69]).
Proposition 2.11. Let S be a semigroup on X. If E is a nonempty subset of X and
there exists τ ∈ G+ such that γτ+ (E) is relatively compact, then ω(E) is nonempty,
compact and attracts E.
In the case X = Rn , the hypotheses of Proposition 2.11 hold if we only assume
that γτ+ (E) is bounded. If there exists t0 ∈ G+ such that S(t) is compact for t > t0 in
G+ , then the hypotheses of Proposition 2.11 still hold, when γτ+ (E) is bounded, even if
X is infinite-dimensional. Semigroups that are compact for t > 0 occur in the study of
parabolic equations or retarded differential equations etc...However, there are examples,
like the damped wave equation, where the associated semigroup is not compact and yet
the properties given in Proposition 2.11 hold. For this reason, we consider the more
general class of asymptotically smooth semigroups, which has been introduced in 1972
by Hale, LaSalle and Slemrod [HaLaSSl].
Definition 2.12. The semigroup S is asymptotically smooth if, for any nonempty,
closed, bounded set B ⊂ X, there exists a nonempty compact set J = J(B) such that
J attracts {x ∈ B | S(t)x ∈ B, ∀ t ∈ G+ }.
One remarks that S is asymptotically smooth if and only if, for any nonempty,
- 14 closed, bounded set B ⊂ X for which S(t)B ⊂ B, for any t ∈ G+ , there exists a
compact set J ⊂ B such that J attracts B. (see [Hal88]).
Obviously, if the semigroup S(t) is compact for t > t0 ≥ 0, then S(t) is asymptotically smooth. Asymptotically smooth semigroups have the following important property
(see [Hal88, Chapters 2 and 3])
Proposition 2.13. If S is an asymptotically smooth semigroup on X and E is a
nonempty subset of X such that γτ+ (E) is bounded for some τ ∈ G+ , then ω(E) is
nonempty, compact, invariant and attracts E.
In 1987, Ladyshenskaya [La87a] introduced the notion of asymptotically compact
semigroups:
Definition 2.14. The semigroup S is asymptotically compact if, for any bounded
subset B of X such that γτ+ (B) is bounded for some τ ∈ G+ , every set of the form
{S(tn )zn }, with zn ∈ B and tn ∈ G+ , tn →n→+∞ +∞, tn ≥ τ , is relatively compact.
We remark that Proposition 2.13 at once implies that every asymptotically smooth
semigroup S is asymptotically compact. On the other hand, Ladyshenskaya [La87a]
had proved that, if S(t) is an asymptotically compact semigroup on X and E is a
nonempty subset of X such that γτ+ (E) is bounded for some τ ∈ G+ , then ω(E) is
nonempty, compact and attracts E. From this result, one immediately deduces that
any asymptotically compact semigroup is asymptotically smooth, obtaining thus the
following result.
Proposition 2.15. Let S be a semigroup on X. Then, S is asymptotically smooth if
and only if it is asymptotically compact.
Since the concepts of asymptotically compact and asymptotically smooth are equivalent, I will not distinguish them in the sequel. I prefer to use the term asymptotically
smooth, because it appeared first. Moreover, the term asymptotically compact is now
misleading, because some authors, like Ball [Ba2], Sell and You [SeYou], call a semigroup asymptotically compact if for any bounded subset B of X, any set of the form
{S(tn )zn }, with zn ∈ B and tn ∈ R, tn →n→+∞ +∞, tn ≥ τ , is relatively compact. The
property of eventual boundedness of orbits of bounded sets is included in this definition,
whereas, this is not the case for asymptotically smooth semigroups.
- 15 In 1982, in his study of the homotopy index for semiflows in non locally compact
spaces, Rybakowski introduced the related concept of admissibility (see [Ry82] and also
the appendix of [HMO]). Let S(t) be a (local) continuous semigroup on X and N be a
closed subset of X. The subset N is called S-admissible if for every sequence {zn } ⊂ N
and every sequence tn ∈ R, tn →n→+∞ +∞ such that {S(t)zn | t ∈ [0, tn ]} ⊂ N , for any
n ∈ N, the set {S(tn )zn | n ∈ N} is relatively compact. As pointed out by Rybakowski,
the notions of admissibility and asymptotically smooth semigroups are not equivalent
[HMO].
Remark 2.16.
(i) One can also show that, if S(t) is an asymptotically smooth semigroup on X and E
is a nonempty subset of X such that Γ− (E) is nonempty and Γ(E) is bounded, then
α(E) is nonempty, compact, invariant, and δX (H(t, E), α(E)) →t→+∞ 0 (for a proof,
see [Hal88, Chapters 2 and 3] or [GR00]).
(ii) Likewise, one shows that, if S(t) is asymptotically smooth and there exists a bounded
complete orbit uz ∈ C 0 (R, X) through some z ∈ X, then the αuz -limit set αuz (z)
is nonempty, compact, invariant and δX (uz (−t), αuz (z)) →t→+∞ 0. Statement (ii)
of Remarks 2.10 implies that αuz (z) is invariantly connected. If, moreover S(t) is a
continuous semigroup, then αuz (z) is connected.
Proposition 2.13 indicates that, if a global attractor A exists, then A contains the
ω-limit set of any bounded set.
2.3. Global attractors
We are now ready to recall the definition of a global attractor and state its basic properties. In this paragraph, we also give the fundamental theorem of existence of compact
global attractors.
Definition 2.17. A nonempty subset A of X is called a global attractor of the semigroup S if
1) A is a closed, bounded subset of X,
2) A is invariant under the semigroup S,
3) A attracts every bounded subset B of X under the semigroup S.
In the same way, one defines local attractors. A nonempty subset J of X is a local
attractor if J is closed, bounded, invariant and attracts a neighbourhood of itself. In
- 16 the past, a special class of global attractors has mainly been studied: it is the class of
compact global attractors.
If S(t) is a continuous semigroup and the global attractor A is compact, it is
straightforward to show that, given a trajectory γ + (x0 ), there exist sequences of positive
numbers εn , tn and a sequence of points yn ∈ A, such that εn →n→+∞ 0, tn+1 > tn ,
tn+1 − tn →n→+∞ +∞, and δX (S(t)x0 , S(t − tn )yn ) ≤ εn , for any tn ≤ t ≤ tn+1 .
Moreover, δX (yn+1 , S(tn+1 − tn )yn ) decays to 0.
We also remark that if A is the global attractor of a semigroup S(t), t ∈ R, then,
for any t0 > 0, A is the global attractor of the discrete semigroup generated by S(t0 ).
Conversely, if S(t0 ) has a compact global attractor A0 and that S(t) is a continuous
semigroup, then A0 is also the global attractor of S(t), t ∈ R.
Before giving the main theorem of existence of global attractors, we describe some
fundamental properties of the global (and local) attractors.
Properties of global attractors.
The following properties of a global attractor are a direct consequence of its definition and of Lemma 2.4.
Lemma 2.18. If the semigroup S admits a global attractor A, the following properties
hold:
a) If B is an invariant bounded subset of X, then B ⊂ A (maximality property).
b) If B is a closed subset of X, which attracts every bounded subset B of X, then
A ⊂ B (minimality property).
c) A is unique.
d) A = {bounded complete orbits of S(t)}.
In the case of a Banach space X, as an immediate consequence of Lemma 2.9 2),
we have the following connectedness result, due to Massat [Ma83a]:
Proposition 2.19. Let S be a semigroup on a Banach space X and A be a compact,
invariant set attracting any compact set of X, then A is connected. In particular, if S
has a compact global attractor A, then A is connected.
Even when X is not a Banach space, we obtain some connectedness properties of
the global attractor. The next proposition generalizes Theorem 4.2 of [Ba2] as well as
- 17 Theorem 3.1 of [GoSa].
Proposition 2.20. If S is a semigroup on a connected metric space X and if A is a
compact, invariant set attracting any compact set of X, A is invariantly connected. If,
in addition, S(t) is a continuous semigroup on X, then A is connected.
Proof. We begin by showing by contradiction that if A is a compact, invariant set
attracting any compact set of X, A is invariantly connected. To simplify the notation, we
assume, without loss of generality, that S is a discrete semigroup. If A is not invariantly
connected, then A = A1 ∪ A2 , where A1 , A2 are nonempty, disjoint, compact, positively
invariant subsets of X. We fix ε > 0 so that NX (A1 , ε) ∩ NX (A2 , ε) = ∅. Since A1 and
A2 are invariantly connected, the continuity of the mapping S implies that there exists
δ, 0 < δ < ε/2, such that S(N X (Ai , δ)) ⊂ NX (Ai , ε), for i = 1, 2. We set, for i = 1, 2,
Xi = {x ∈ X | there exists n0 = n0 (x) such that S(n)x ∈ NX (Ai , δ), n ≥ n0 }
Clearly, X1 ∩ X2 = ∅. Moreover, the following properties hold:
(i) X = X1 ∪ X2 . Indeed, let x ∈ X. As x is attracted by A, there exists n0 (x) ≡ n0
such that {S(n)x | n ≥ n0 } ⊂ NX (A, δ). Assume that S(n0 )x ∈ NX (A1 , δ). Then,
S(n0 + 1)x ∈ NX (A, δ) ∩ NX (A1 , ε), which implies that S(n0 + 1)x ∈ NX (A1 , δ). By
recursion, it follows that S(n)x ∈ NX (A1 , δ), for n ≥ n0 and thus x ∈ X1 .
(ii) Xi 6= ∅, for i = 1, 2. Indeed, due to the positive invariance of Ai , we have the
inclusion Ai ⊂ Xi .
(iii) X1 and X2 are closed sets. If x1 ∈ X1 , there exist ym ∈ X1 , m ∈ N, such that ym
converges to x1 as m goes to ∞. Since A attracts every compact set of X, there exists
n0 > 0 such that {S(n)K | n ≥ n0 } ⊂ NX (A, δ), where K = {x1 } ∪ {ym | m ∈ N}. The
arguments of (i) imply that S(n)ym ∈ NX (A1 , δ), for n ≥ n0 and m ∈ N. Due to the
continuity of the map S(n0 ), there exists m0 > 0 such that d(S(n0 )ym , S(n0 )x1 ) ≤ δ,
for m ≥ m0 . It follows that x1 ∈ NX (A1 , 2δ) ⊂ NX (A1 , ε). Hence, {S(n)x1 | n ≥ n0 } ⊂
NX (A1 , δ) and x1 ∈ X1 .
We have thus proved that the connected set X is the union of the two closed, disjoint compact subsets X1 and X2 , which is a contradiction. Therefore A is invariantly
connected.
To prove that A is connected when, in addition, S(t) is a continuous semigroup, we
argue by contradiction and follow the lines of the above proof. If A is not connected,
- 18 then A = A1 ∪ A2 , where A1 , A2 are disjoint, nonempty, compact sets. Like before, one
introduces two positive constants ε and δ and, for i = 1, 2, the set
Xi = {x ∈ X | there exists τ0 = τ0 (x) such that S(t)x ∈ NX (Ai , δ), t ≥ τ0 }
Clearly X1 ∩ X2 = ∅ and the following properties hold:
(a) X = X1 ∪ X2 . Let x ∈ X. Since x is attracted by A, there exists τ > 0 such
that {S(t)x | t ≥ τ } ⊂ NX (A, δ). As S(t) is a continuous semigroup, {S(t)x | t ≥ τ } is
connected and therefore is either completely contained in NX (A1 , ε) or NX (A2 , ε).
(b) Ai ⊂ Xi , for i = 1, 2. Indeed, let a ∈ Ai . Due to the invariance of A, there exists
b ∈ A and T > 0 such that S(T )b = a and S(t + T )b = S(t)a ∈ A, for any t ≥ 0. Since
S(t) is a continuous semigroup, {S(t)a = S(T + t)b | t ≥ 0} is connected and therefore
completely contained in Ai .
(c) One shows like in (iii) that X1 and X2 are closed subsets of X.
Thus, X is the union of the two closed, disjoint compact subsets X1 and X2 , which is
a contradiction. Hence A is connected.
Remarks.
1. As an immediate consequence of Proposition 2.20, we notice that if the set A satisfies
the hypotheses of Proposition 2.20, if, for any x ∈ X, the map t ∈ [0, +∞) 7→ S(t)x ∈ X
is continuous and if X is not connected, then every connected component of X contains
exactly one connected component of A.
2. If S(t) is not a continuous semigroup on X, then, in general, under the hypotheses
of Proposition 2.20, the set A is only invariantly connected. Gobbino and Sardella
[GoSa] give an example of a discrete semigroup defined on a connected metric space X
which has a non connected compact global attractor. Modifying the proof of [GoSa,
Proposition 4.3] in the above way, one shows that, if S(t) is any semigroup on X and
if A is a compact, invariant set attracting every compact set of X, then either A is
connected or A has infinitely many connected components. This fact directly implies
that, when X is connected and locally connected, the compact global attractor A is
connected.
Before giving the fundamental theorem of existence of compact global attractors,
we introduce the important notions of stability and dissipativness.
Let S be a semigroup on the metric space X. A set J ⊂ X attracts points locally
if there exists a neighbourhood U of J such that J attracts each point of U under S.
- 19 We recall that E ⊂ X is (Lyapunov) stable if, for any neighbourhood V of E, there
exists a neighbourhood W ⊂ V of E such that
S(t)W ⊂ V ,
∀t ∈ G+ .
(2.10)
We say that E is stable for t ≥ τ in G+ if the above inclusion (2.10) holds only for any
t ∈ G+ , t ≥ τ . The set E is asymptotically stable if E is stable and attracts points
locally. The set E is uniformly asymptotically stable if it is asymptotically stable and
attracts a neighbourhood of itself.
The next theorem describes a stability result for compact global attractors.
Theorem 2.21. Let S be a semigroup on X. If A is a compact, positively invariant
set, which attracts a neighbourhood of itself, the following properties hold:
(i) if the mapping (t, z) ∈ G+ × X 7→ S(t)z ∈ X is continuous, then A is stable and thus
uniformly asymptotically stable. In particular, A is uniformly asymptotically stable if
S is a discrete semigroup.
(ii) if S(t) is a continuous semigroup, then, for any τ > 0, A is stable for t ≥ τ .
Proof. (i) Let ε > 0 be a fixed positive number. By assumption, there exists a neighbourhood W1 of A and t1 ∈ G+ such that, for any t ∈ G+ , t ≥ t1 ,
S(t)W1 ⊂ NX (A, ε) .
Moreover, the joint continuity of the mapping (t, z) ∈ G+ ×X 7→ S(t)z ∈ X implies that,
for any a0i ∈ A, there exists η(a0i ) > 0 such that, for any z ∈ X with δX (z, a0i ) < η(a0i )
and any t ∈ G+ ∩ [0, t1 ],
δX (S(t)z, S(t)a0i) < ε ,
and hence, due to the positive invariance of A
S(t)z ∈ NX (S(t)A, ε) ⊂ NX (A, ε) .
(2.11)
Since the set A is compact, it can be covered by a finite number k of neighbourhoods
NX (a0i , η(a0i )). Hence, there exists a positive number η such that A ⊂ NX (A, η) ⊂
i=k
∪i=1
NX (a0i , η(a0i )). We deduce from (2.11) that
S(t)NX (A, η) ⊂ NX (A, ε) ,
∀t ∈ G+ ∩ [0, t1 ] .
- 20 If we set W = W1 ∩ NX (A, η), then
S(t)W ⊂ NX (A, ε) ,
∀t ∈ G+ .
We finally remark that, for any discrete dynamical system, the mapping (t, z) ∈ N ×
X 7→ S(t)z ∈ X is continuous.
(ii) We recall that, due to a result of [CM], if S(t) is a continuous semigroup, the mapping
(t, z) ∈ [τ, +∞) × X 7→ S(t)z ∈ X is continuous, for any τ > 0. The arguments used in
(i) then show that A is stable for t ≥ τ .
Other results involving stability properties are given in [Hal88].
We finally introduce the concept of dissipativness.
Definition 2.22. The semigroup S is point (compact) (locally compact) (bounded)
dissipative on X if there exists a bounded set B0 ⊂ X, which attracts each point
(compact set) (a neighbourhood of each compact set) (bounded set) of X.
If the semigroup is bounded dissipative, there exists a bounded set B1 ⊂ X with
the property that, for any bounded set B ⊂ X, there exists τ = τ (B) ∈ G+ such that
γτ+ (B) ⊂ B1 . Such a set B1 is called an absorbing set for S.
If S is a semigroup on X, which admits a global attractor, then S is bounded dissipative. Of course, if S is bounded dissipative, it has not necessarily a global attractor,
as it is easily seen in Example 2.8, where the unit ball in H0 is an absorbing set and
nevertheless there does not exist a global attractor.
An interesting implication of dissipativness for asymptotically smooth semigroups
is as follows ([Hal00], [ChHa]):
Theorem 2.23. Let F be a family of subsets of X. If S is an asymptotically smooth
semigroup on X and there is a bounded set B in X that attracts each element of F ,
then there exists a compact invariant set which attracts every element of F .
Example 2.24. The notions of point dissipative, compact dissipative and bounded
dissipative are not equivalent in general, as is shown in this example, which is a modification of an unpublished example of the thesis of Cooperman ([Coo]) and is described
in [ChHa].
Let H = l2 be the Hilbert space of square summable series {x = (x1 , x2 , . . .) | xi ∈ R , i =
- 21 P∞
1, 2, . . . , kxk2 ≡ i=1 |xi |2 < ∞} with the orthonormal basis ej = (0, . . . , 0, 1, 0, . . .),
j = 1, 2, . . ., where the number 1 appears at the j th position. Also, we denote by θ the
zero element of l2 and introduce the following points
x1,j =
1
ej ,
2
j≥2,
and xn,j =
1
ej + 2n−2 e1 ,
2n
j≥n≥2,
We define an auxiliary map T : H → H by first setting
T (xn,j ) =
1
ej + 2n e1 ,
2n
n≥1,
j ≥ max(n, 2) .
(2.12)
Then, we extend T to a map from H to H as follows.
If x belongs to the ball BH (xn,j , cn,j ) of center xn,j and radius cn,j , n ≥ 1, j ≥ max(n, 2),
where cn,j = 14 21j , we set, using (2.12):
T (x) =
kx − xn,j kxn,j + cn,j (x − xn,j ) + (cn,j − kx − xn,j k)T (xn,j )
,
cn,j
otherwise, we simply set T (x) = x.
We finally introduce the mapping S : H → H given by
1
S(x) = T ( x) ,
2
x∈H .
Cholewa and Hale [ChHa] have shown that S : H → H is continuous and asymptotically
smooth. Moreover, each point of H has a neighbourhood which is attracted by the zero
element θ in l2 , which implies that S is compact dissipative. Yet, there is no bounded
set in H attracting each bounded set of H. Indeed, one at once proves by recursion that
S j (ej ) =
1
ej + 2j e1 ,
2j
j≥2,
which implies that, for each k ≥ 1, the set γk+ ({x ∈ H | kxk = 1}) is unbounded in H.
However, the different notions of dissipativness can be equivalent in some cases. For
example, it has been proved by Massat ( [Ma83a], see also [HVL]) that point dissipativness and compact dissipativness are equivalent for certain classes of neutral functional
differential equations which arise in connection with the telegraph equation. More generally, if the semigroup S is asymptotically smooth, one can relate the different types
of dissipativeness. Assume that S is asymptotically smooth and point dissipative. If,
for any compact subset K of X, there exists τ ≥ 0 such that γτ+ (K) is bounded, then
- 22 S is locally compact dissipative. If, for any bounded subset B of X, there exists τ ≥ 0
such that γτ+ (B) is bounded, then S is bounded dissipative. (see [Ma83a] and also
[Hal88]). The existence of compact global attractors is actually a consequence of the
latter property and of Theorem 2.23, although we shall give a different proof below.
Existence of global attractors.
Let us recall that a set A is said to be a maximal compact invariant set if it is a
compact invariant set under S and is maximal with respect to these properties. One
of the basic results concerning the existence of a maximal compact invariant set (see
[HaLaSSl]) is the following:
Theorem 2.25. If S is a semigroup on X such that there is a nonempty compact set
K that attracts each compact set of X, and A = ∩t∈G+ S(t)K, then
(i) A is independent of K,
(ii) A is the maximal compact invariant set in X,
(iii) A attracts compact sets.
And the connectedness properties given in Proposition 2.19 and Proposition 2.20 hold.
Remark. The above set A is invariant and attracts compact sets. It is expected to be
the compact global attractor. Unfortunately, without additional hypotheses, this is not
the case, as we have already seen in Example 2.24.
A simpler example where A satisfies the properties of Theorem 2.25 and is not a compact global attractor is described in [Hal99]. Let H be a separable Hilbert space with
orthonormal basis ej , j ≥ 1 and let λj , j ≥ 1 be a sequence of real numbers, 0 < λj < 1,
for j ≥ 1. We introduce the linear mapping defined on the basis vectors by Sej = λj ej .
Since kSxkH ≤ kxkH , for all x ∈ H, γ + (B) is bounded if B is bounded. Clearly,
S n x → 0 as n → +∞, for any x ∈ H; and {0} attracts a neighbourhood of every point
and thus every compact set. If λj ≤ λ < 1, then {0} is the global attractor of S. But,
if, for instance, λj → 1, when j → +∞, then {0} cannot be a global attractor. One
remarks that, in the first case, the radius r(σ(S)) of the spectrum of S is strictly less
than 1, while in the second case r(σ(S)) = r(σess (S)) is equal to 1, where r(σess (S)) is
the radius of the essential spectrum of S. This example is actually a simple illustration
of the general theorem (see [Hal99] and also [Hal88, Section 2.3]), which states that if
S is a bounded linear point dissipative map on a Banah space X, then {0} attracts
compact sets. It is the compact global attractor if and only if r(σ(S)) < 1 or also if and
- 23 only if r(σess (S)) < 1.
We now state and prove the fundamental theorem of existence of a compact global
attractor. A first version of it is due to [HaLaSSl]. Several different proofs of the
statements below can be found in [Hal85, Theorem 2.2], [Hal88, Theorem 2.4.6], [La87a,
Theorem3.1] (see also [Te] and, for a generalization to multivalued maps, [Ba2]). Here
we give a simple proof, using mainly Proposition 2.13.
Theorem 2.26. (Existence of a compact global attractor) The semigroup S(t), t ∈ G+ ,
on X admits a compact global attractor A in X if and only if
(i) S(t) is asymptotically smooth,
(ii) S(t) is point dissipative,
(iii) For any bounded set B ⊂ X, there exists τ ∈ G+ such that γτ+ (B) is bounded.
Moreover,
[
A =
{ω(B) | B bounded subset of X} .
(2.13)
And the connectedness properties given in Proposition 2.19 and Proposition 2.20 hold.
In the proof of Theorem 2.26, we use the following auxiliary result.
Lemma 2.27. Assume that the semigroup S(t), t ∈ G+ , on X is point dissipative and
that, for any bounded set B ⊂ X, there exists τ ∈ G+ such that γτ+ (B) is bounded.
Then, there is a bounded set B1 ⊂ X such that, for any compact subset K of X, there
exist ε = ε(K) > 0 and t1 = t1 (K) ∈ G+ such that
S(t)(NX (K, ε)) ⊂ B1 ,
∀t ≥ t1 (K) ,
t ∈ G+ .
(2.14)
Proof. Without loss of generality, we may assume that G+ = [0, +∞). Since S(t) is
point dissipative, there exists a bounded set B0 , which may be assumed to be open,
such that, for any x0 ∈ X, there is a time t∗ (x0 ) ≥ 0 with
S(t)x0 ⊂ B0 ,
∀t ≥ t∗ (x0 ) .
As S(t∗ (x0 )) is continuous from X into X, we can find ε(x0 ) > 0 such that
S(t∗ (x0 ))(NX (x0 , ε(x0 ))) ⊂ B0 ,
and thus, for s ≥ τ0 , where τ0 is chosen so that γτ+0 (B0 ) is bounded,
S(s + t∗ (x0 ))(NX (x0 , ε(x0 ))) ⊂ γτ+0 (B0 ) ≡ B1 .
- 24 If K is a compact set in X, we can cover K by a finite number of neighbourhoods
NX (x0i , ε(x0i )), 1 ≤ i ≤ k, where x0i ∈ K. Moreover, there exists ε(K) > 0, such that
Si=k
K ⊂ NX (K, ε(K)) ⊂ i=1 NX (x0i , ε(x0i )). Finally, if we set t1 (K) = max1≤i≤k (τ0 +
t∗ (x0i )), we have
S(t)(NX (K, ε(K))) ⊂ B1 , ∀t ≥ t1 .
(2.15)
Proof of Theorem 2.26 Clearly, if A is a compact global attractor, then the properties
(i), (ii), (iii) hold.
Conversely, we assume now that the properties (i), (ii), (iii) are satisfied; without loss of
generality, we may also suppose that G+ = [0, +∞). Let B1 be the bounded set which
has been constructed in Lemma 2.27. We set A = ω(B1 ). Due to Proposition 2.13 and
Lemma 2.9, A is nonempty, compact, invariant and attracts B1 . Actually, A attracts
any bounded set B of X. Indeed, again according to Proposition 2.13, the ω-limit set
K = ω(B) is nonempty, compact and attracts B. Let ε be a real number such that
0 < ε < ε(K), where ε(K) has been introduced in Lemma 2.27. Since K attracts B,
there exists t0 > 0 such that
S(t)B ⊂ NX (K, ε) ,
t ≥ t0 ,
and therefore, for t ≥ 0,
S(t)S(t1 (K) + t0 )B ⊂ S(t)S(t1 (K))NX (K, ε) ⊂ S(t)B1 ,
where t1 (K) has been defined in Lemma 2.27. Since A attracts B1 , it follows that A
also attracts B.
Due to Lemma 2.18, A contains any bounded invariant set and, in particular, the ω-limit
set ω(B) of any bounded set B. Since A = ω(A), the equality (2.13) holds.
Example 2.28. The existence of a global attractor does not necessarily imply that,
for any bounded set B ⊂ X, the orbit γ + (B) is bounded. For instance, if S(t) is a
continuous semigroup, which is not continuous at t = 0, then, due to the lack of continuity at t = 0, the size of γτ+ (B) can grow to ∞, when τ → 0; such a semigroup
generated by an evolutionary equation will be introduced in Section 4.5. Here we construct a continuous map S : l2 → l2 , which has a compact global attractor and yet the
image through S of the unit ball is unbounded. Like before, we consider the Hilbert
space H = l2 of square summable series, with the orthonormal basis ej , j = 1, 2, . . ..
- 25 Let ϕ : s ∈ [0, +∞] 7→ ϕ(s) ∈ [0, 1] be a continuous function such that ϕ(s) = 1, if
0 ≤ s ≤ 81 and ϕ(s) = 0, if s ≥ 41 . We define the map S in the following way:
1
S(x) = jϕ(k ej − xk)e1 ,
2
S(x) = θ , otherwise ,
1
1
∀x ∈ BH ( ej , ) ,
2
4
j≥2,
where θ is the zero element of l2 . Clearly, S k (x) = θ, for any x ∈ H and k ≥ 2; thus
{θ} is the compact global attractor. On the other hand, S(BH (θ, 1)) is unbounded in
H.
We recall that an equilibrium point of the semigroup S(t) is a point x ∈ X such
that S(t)x = x, for any t ∈ G+ .
The following result had already been proved by Billotti and LaSalle in 1971 (see
[BiLaS]). Here, we deduce it from Theorem 2.26.
Theorem 2.29. Assume that S(t) is either a continuous semigroup or a discrete semigroup and that S is point dissipative. If there exists t1 ∈ G+ such that the semigroup
S(t) is compact for t ∈ G+ , t > t1 , then there exists a compact global attractor A in
X. Moreover, if X is a Banach space and t1 = 0, there exists (at least) an equilibrium
point of S(t).
Proof. We give the proof in the case where G+ = [0, +∞). Theorem 2.29 is a direct consequence of Theorem 2.26, if we show that, for any bounded set B ⊂ X,
there exists τ ≥ 0 such that γτ+ (B) is bounded. Since S(t)B is relatively compact,
for t > t1 , it is sufficient to show this property for any compact set. As S(t) is point
dissipative, there exists a bounded set B ∗ , which may be assumed to be open, which
attracts every point of X. Let t2 = t1 + 2 and B0 = NX (B ∗ , ε0 ), where ε0 > 0. Arguing as in the proof of Lemma 2.27, one shows that, for any compact subset H of
X, there exists t3 (H) > t2 such that S(t)(H) ⊂ γ + (S(t2 )(B0 )), for any t ≥ t3 (H).
Thus, it remains to show that γ + (S(t2 )(B0 )) is bounded. Let K0 = S(t2 )B0 ; as in the
proof of Lemma 2.27, for any x0 ∈ K0 , there are ε(x0 ) > 0 and t(x0 ) > 0 such that
S(t(x0 ))NX (x0 , ε(x0 )) ⊂ B0 . Since K0 is compact, we can cover it by a finite number k
of neighbourhoods NX (x0i , ε(x0i )), 1 ≤ i ≤ k. We set t3 (K0 ) = max1≤i≤k (t2 + t(x0i ))
Ss=t (K )+1
and K3 = s=13 0
S(s)S(t1 + 1)B0 . If S(t) is a continuous semigroup on X, then
K3 is a compact set. Morover, one at once checks that S(s)S(t2 )B0 ⊂ K3 , for any
- 26 s ≥ 0. The last statement of the theorem can be found in [GeKr] and in [BiLaS] (see
also [HaLo]).
The proof of the above theorem is similar when S is a discrete semigroup.
Remark 2.30. Often, one can define a global attractor in a weak sense, when the
semigroup is bounded dissipative. For example, let X be a separable, reflexive Banach
space; we denote by Xw the space X endowed with the corresponding weak topology.
Assume that S(t) is a bounded dissipative continuous semigroup, such that, for each
fixed t, S(t) : Xw → Xw is continuous. Then there is a positive number r such that the
ball B0 = BX (0, r) of center 0 and radius r is an absorbing set for S(t). In particular,
there exists τ ≥ 0 such that S(t)B0 ⊂ B0 , for t ≥ τ . Since X is reflexive and separable,
weak
the weak topology on B0
is metrizable. We denote by dw the corresponding metric.
weak
is compact for the weak topology, the semigroup S(t)Y restricted to
Since B0
weak
is obviously compact on Y for the weak topology and point dissipative.
Y = γτ+ B0
Therefore, by Theorem 2.29, there exists a global attractor A, bounded in X, compact
in Xw , invariant, such that, for any bounded set B ⊂ X,
lim δw (S(t)B, A) = 0 .
t→+∞
2.4. Examples of asymptotically smooth semigroups
We now give some examples of asymptotically smooth semigroups. The motivation
of the hypotheses in the first example probably comes from the Duhamel formula for
evolutionary equations.
If B is a bounded subset of a Banach space X, we set kBkX = supb∈B kbkX .
Theorem 2.31. Let X be a Banach space and S(t), t ∈ G+ , be a semigroup defined
on a closed positively invariant subset M of X. Assume that one can write, for any
t ∈ G+ and any u ∈ M ,
S(t)u = U (t)u + V (t)u ,
(2.16)
where U (t) and V (t) are mappings of M into X, with the property that, for any bounded
set B ⊂ M , there exists τ0 (B) ∈ G+ such that,
(i) U (t)B is relatively compact for any t in G+ , t > τ0 (B),
and
(ii) for any t in G+ , t > τ0 (B),
kV (t)ukX ≤ k(t, kBkX ) ,
∀u ∈ B,
(2.17)
- 27 where k : (s, r) ∈ [0, +∞) × [0, +∞) 7→ [0, +∞) is a function such that k(s, r) → 0 as
s → +∞.
Then S(t) is asymptotically smooth.
Conversely, if a semigroup S(t), t ∈ G+ , admits a compact global attractor A on
a Banach space X, then S(t) must have the representation (2.16) with U (t), V (t)
satisfying (i) and (ii).
Proof. To simplify the notation, we assume that G+ = [0, +∞). Let B be a bounded
subset of M and τ = τ (B) ≥ 0 such that γτ+ (B) is bounded. We consider a set of the
form {S(tn )zn }, with zn ∈ B and tn ∈ [τ, +∞), tn →n→+∞ +∞. It suffices to show
that, for any ε > 0, we can cover the set {S(tn )zn } by a finite number of balls of radius
r ≤ ε. Due to the condition (2.17), there exists t1 = t1 (ε, B) > 0 such that
kV (t)ukX ≤ k(t, kγτ+ (B)kX ) ≤ ε/2 ,
∀u ∈ γτ+ (B) ,
∀t ≥ t1 .
(2.18)
We set t2 = τ0 (γτ+ (B)) and choose t3 > sup(t1 , t2 ). Let n1 ∈ N be such that, for n ≥ n1 ,
tn > τ + t3 ; clearly the set {S(tn )zn | n ≤ n1 } can be covered by a finite number of balls
of radius ε. It remains to show that the set B1 = {S(tn )y | y ∈ B, tn ≥ τ + t3 } can be
covered by a finite number of balls of radius r ≤ ε. Each element of B1 can be written
as
S(t3 )S(tn − t3 )y = U (t3 )S(tn − t3 )y + V (t3 )S(tn − t3 )y .
But U (t3 )S(tn −t3 )y ⊂ U (t3 )γτ+ (B), which is compact and thus can be covered by a finite
number of balls of radius ε/2. Furthermore, due to (2.18), kV (t3 )S(tn − t3 )ykX ≤ ε/2,
for any n ≥ n1 . Thus B1 can be covered by a finite number of balls of radius ε.
Conversely, assume that the semigroup S(t) admits a compact global attractor A on
the Banach space X. Then, due to the compactness of A, for any z ∈ X, there exists at
least one element a ∈ A such that dX (z, a) = δX (z, A); we thus choose such an element
a and denote it by a = P z. The mappings U (t)u = P S(t)u and V (t)u = (Id − P )S(t)u
clearly satisfy the conditions (i) and (ii) of Theorem 2.31.
Remark 2.32. In the case of a Hilbert space (see [Te, Remark I.1.5]) and more generally, in the case of a uniformly convex Banach space, we can choose U (t)u = P0 S(t)u
and V (t)u = (Id − P0 )S(t)u, where P0 is the projection onto the closed convex hull
co(A) of the global attractor A. In this case, U (t) and V (t) are continuous functions of
u.
- 28 Remark 2.33. The following typical example of semigroup satisfying the hypotheses
of Theorem 2.31 is often encountered in the study of semilinear equations. It has been
first studied by Webb ([Web79a]). Let S(t), t ≥ 0, be a continuous semigroup on a
Banach space X such that, for t ≥ 0,
S(t)u = Σ(t)u +
Z
0
t
Σ(t − s)K(S(s)u) ds ,
for any u ∈ M , where M is a closed positively invariant subset of X, Σ(t) is a C 0 semigroup of linear mappings from X into X and K is a compact map from X into X.
We assume that Σ(t) = Σ1 (t) + Σ2 (t), t ≥ 0, where Σ1 (t) is a compact linear map for
t > t0 ≥ 0 and the linear map Σ2 (t) satisfies
kΣ2 (t)kL(X,X) ≤ k2 (t) ,
(2.19)
with k2 : t ∈ [0, +∞) 7→ [0, +∞) is a function such that k2 (t) → 0 as t → +∞.
If the positive orbit of any bounded subset B of M is bounded, one easily shows
([Web79a], [Web79b], [Hal88]) that then S(t) is written as a sum S(t)u = U (t)u+V (t)u,
with U , V satisfying the hypotheses of Theorem 2.31.
In the applications, it is often difficult to determine the decomposition U (t) + V (t)
given in Theorem 2.31. For this reason, we shall give other criteria of asymptotical
smoothness. The following result is due to Ceron and Lopes (see [CeLo]). We recall
that a pseudometric ρ(., .) is precompact (with respect to the norm of X), if any bounded
sequence in X has a subsequence which is a Cauchy sequence with respect to ρ.
Proposition 2.34. Let X be a Banach space and S(t), t ∈ G+ , be a semigroup defined
on a closed positively invariant subset M of X. Assume that, for any bounded set
B ⊂ M , there exists τ0 (B) in G+ , such that, for any u1 , u2 in B and for any t in G+ ,
t ≥ τ0 (B),
kS(t)u1 − S(t)u2 kX ≤ k(t, kBkX )ku1 − u2 kX + ρt,kBkX (u1 , u2 ) ,
(2.20)
where k : (s, r) ∈ [0, +∞) × [0, +∞) 7→ [0, +∞) is a function such that k(s, r) → 0 as
s → +∞ and ρt,kBkX is a precompact pseudometric, for t in G+ , t > τ0 (B).
Then S(t) is asymptotically smooth.
The proof of Proposition 2.34 is very similar to the one of Theorem 2.31.
- 29 We now give a third criterium of asymptotical smoothness, which deals with functionals and has first been introduced by J. Ball ([Ba1]) and then applied to dispersive
equations by Abounouh ([Ab1]), Ghidaglia ([Gh94]) etc....
Proposition 2.35. Let Y be a topological space and X be a uniformly convex Banach
space such that X is continuously embedded into Y . We consider a semigroup S(t),
t ≥ 0, on X satisfying the following properties:
(i) for any t ≥ 0, the mapping S(t) is continuous on the bounded subsets of X, for the
topology of Y ;
(ii) for any bounded set B of X such that γτ+ (B) is bounded in X for some τ ≥ 0, every
sequence S(tj )bj , where bj ∈ B and tj →j→∞ +∞, is relatively compact in Y ;
(iii) for any x0 ∈ X and t ≥ 0, we have
Z t
F (S(t)x0 ) = exp(−γt)F (x0 ) +
exp(−γ(t − s))F1 (S(s)x0 ) ds ,
(2.21)
0
kxkpX
+ F0 (x), p > 0 and F0 , F1 are continuous functionals on the
where γ > 0, F (x) =
bounded sets of X for the topology of Y and are bounded on the bounded sets of X.
Then the semigroup S(t) is asymptotically smooth in X.
Proof. We recall that any uniformly convex Banach space is reflexive. Let B be a
bounded set in X such that γτ+ (B) is bounded in X for some τ ≥ 0 and let S(tj )bj be
a sequence such that bj ∈ B and tj →j→∞ +∞. Since X is reflexive, there exists a
subsequence, still labelled by j, such that S(tj )bj ⇀ z weakly in X, where z ∈ B0 ≡
co(γτ+ (B)), the closed convex hull of γτ+ (B). Due to (ii), we can also suppose that
S(tj )bj → z in Y as j → +∞. We want to show that S(t′j )bj ′ → z in X, where j ′ is
a subsequence of j, j ′ → +∞. Since X is uniformly convex, it suffices to show that
limj ′ →+∞ kS(tj ′ )bj ′ kX = kzkX , for some subsequence j ′ of j. As S(tj )bj converges
weakly to z, we already know that kzkX ≤ lim inf j→+∞ kS(tj )bj kX . Thus, it remains
to show that lim supj ′ →+∞ kS(tj ′ )bj ′ kX ≤ kzkX , for some subsequence j ′ of j.
For each n ∈ N, there exists a subsequence j n such that S(tj n − n)bj n ⇀ zn weakly in
X and S(tj n − n)bj n → zn in Y . By a diagonalization argument, we can construct a
subsequence j ′ such that
S(tj ′ −n)bj ′ ⇀ zn , weakly in X ,
and S(tj ′ −n)bj ′ → zn ∈ Y , ∀ n ∈ N . (2.22)
We infer from (i) that, for any t ≥ 0, S(tj ′ − n + t)bj ′ → S(t)zn in Y . In particular,
S(n)zn = z. We consider now the equality (2.21) for t = n and x0 = S(tj ′ − n)bj ′ , when
- 30 tj ′ − n ≥ τ . From (2.21), (2.22) and the dominated convergence theorem of Lebesgue,
we deduce that
lim sup(kS(tj ′ )bj ′ )kpX ) + F0 (z) ≤ exp(−γn) sup (|F (x)|)
x∈B0
j ′ →+∞
+
Z
0
or also, since F (z) = exp(−γn)F (zn) +
Rn
0
n
exp(−γ(n − s))F1 (S(s)zn ) ds ,
exp(−γ(n − s))F1 (S(s)zn ) ds,
lim sup(kS(tj ′ )bj ′ )kpX ) ≤ 2 exp(−γn) sup (|F (x)|) + kzkpX .
x∈B0
j ′ →+∞
Letting n go to +∞, we obtain that
lim sup kS(tj ′ )bj ′ )kX ≤ kzkX .
j ′ →+∞
The proposition is thus proved.
Remarks 2.36.
(i) In the applications, the space Y is often the space X endowed with the weak topology of X. Then the condition (ii) is always satisfied. In this case, we can also assume
that X is only a reflexive Banach space and replace the conditions on the functional F
by the hypothesis:
(H.1) F : X → R+ is continuous, bounded on the bounded sets of X and the properties S(tj )bj ⇀ z weakly in X, where bj is bounded in X, tj →j→+∞ +∞, and
lim supj→+∞ F (S(tj bj )) ≤ F (z) imply that S(tj )bj converges strongly to z in Z, where
Z is a Banach space such that X ⊂ Z, with continuous injection.
Then, one shows like in the proof of Proposition 2.35 that, if B is a bounded set in
X such that γτ+ (B) is bounded in X for some τ ≥ 0 and if S(tj )bj is a sequence such
that bj ∈ B and tj →j→∞ +∞, then there exists a subsequence j ′ , such that S(tj ′ )bj ′
converges strongly in Z to some element z ∈ B0 ≡ coX (γτ+ (B)).
(ii) Moise, Rosa and Wang [MRW] consider more general functionals on Y , in the case
where Y is the space X endowed with the weak topology.
(iii) A similar result holds for discrete semigroups S provided that the equality (2.21)
is replaced by
n
F (S x0 ) = exp(−γn)F (x0 ) +
n
X
m=0
exp(−γ(n − m))F1 (S m x0 ) ds ,
∀n ∈ N . (2.23)
- 31 Further example of asymptotically smooth semigroups: α-contracting and
condensing semigroups.
Another class of examples of asymptotically smooth semigroups is given by the
α-contracting and condensing semigroups. Let now X be a Banach space and B be the
set of its bounded subsets. The mapping α : B → [0, +∞), defined by
α(B) = inf{l > 0 | B admits a finite cover by sets of diameter ≤ l} ,
is called the (Kuratowski)-measure of noncompactness or, shorter, the α-measure of
noncompactness. It has the following properties (see [De], for example):
(a) α(B) = 0 if and only if B is compact;
(b) α(·) is a seminorm, i.e., α(λB) = |λ|α(B) and α(B1 + B2 ) ≤ α(B1 ) + α(B2 );
(c) B1 ⊂ B2 implies α(B1 ) ≤ α(B2 ); α(B1 ∪ B2 ) = max(α(B1 ), α(B2 ));
(d) α(B) = α(co(B)) ;
(e) α is continuous with respect to the Hausdorff distance HdistX .
A continuous map S : X → X is a conditional α-contraction of order k, 0 ≤ k < 1,
with respect to the measure α if α(S(B)) ≤ kα(B), for all bounded sets B ⊂ X for
which S(B) is bounded. The map S is an α-contraction of order k if it is a conditional
α-contraction of order k and a bounded map. A continuous map S : X → X is a
conditional α-condensing map, with respect to the measure α if α(S(B)) < α(B), for
all bounded sets B ⊂ X for which S(B) is bounded and α(B) > 0. The map S is
α-condensing if it is conditional α-condensing and bounded.
Every bounded linear operator S can be written in the form S = U + V , where the
linear map U is compact and the spectral radius of V is the same as the radius of the essential spectrum of S. Also there exists an equivalent norm on X such that kSkL(X,X) =
r(σess (S)), with respect to this new norm. From a result of Nussbaum [Nu1] stating that
r(σess (S)) = limn→+∞ (α(S n ))1/n , it follows that S is an α-contraction with respect to
a norm equivalent to the one of X if and only if r(σess (S)) < 1.
The prototype of α-contraction is given by the nonlinear map S = U +V , where U is
a nonlinear compact map and V is a globally Lipschitz mapping with Lipschitz constant
k, 0 ≤ k < 1. In this case, S is an α-contraction of order k. If S is a mapping satisfying
the condition (2.20), for any bounded set B ⊂ X, where 0 ≤ k < 1 is independent of B,
S is an α-contraction of order k.
Conditional α-condensing maps are asymptotically smooth (for a proof, see [Ma80],
- 32 [Hal88]). One should notice that asymptotically smooth maps are not necessarily conditional α-condensing, as it is shown by an example in [ChHa, Proposition 1.1]. This fact
is somehow expected since the definition of α-condensing involves the metric whereas
the definition of asymptotically smooth only involves the topology. So we can wonder if
for any asymptotically smooth map S on a Banach space, there exists another equivalent
norm for which S is conditional α-condensing.
The following interesting property also holds:
Theorem 2.37. Let X be a Banach space. If S : X → X is an α-condensing and
compact dissipative mapping, then S has a fixed point.
This theorem was discovered independently by Nussbaum [Nu2] and by Hale and
Lopes [HaLo].
Similar definitions and properties hold for continuous semigroups S(t), t ∈ [0, +∞).
A semigroup S(t) on X is a conditional α-contraction if there exists a continuous function k : [0, +∞) → [0, +∞) such that k(t) → 0 as t → +∞ and, for each t > 0 and
each bounded set B ⊂ X for which S(t)B is bounded, one has α(S(t)B) ≤ k(t)α(B).
The function k(t) is called the contracting function of S(t). The semigroup S(t) is an
α-contraction if it is a conditional α-contraction and, for each t > 0, the set S(t)B is
bounded if B is bounded. Likewise, the semigroup S(t), t ≥ 0, is α-condensing if, for any
bounded set B in X and for any t > 0, the set S(t)B is bounded and α(S(t)B) < α(B)
if α(B) > 0.
It is shown in [CeLo] that if S(t) is a semigroup satisfying the assumptions of Proposition 2.34 with M = X and the function k is independent of kBkX , then S(t) is
an α-contraction. In particular, if S(t) is a semigroup satisfying the assumptions of
Theorem 2.31, where M = X and V (t) is a globally Lipschitz function with Lipschitz
constant k(t) →t→+∞ 0, then S(t) is an α-contraction.
Conditional α-contractions S(t) are asymptotically smooth (see [Hal88]). Theorem 2.37
and Theorem 2.26 imply the following result (for a proof, see [Hal88, Section 3.4]):
Theorem 2.38. Let X be a Banach space and S(t), t ≥ 0, be a continuous semigroup
on X. If moreover S(t), t ≥ 0, is an α-contraction with contracting function k(t) ∈ [0, 1),
is point dissipative and if, for any bounded set B ⊂ X, there exists τ ≥ 0 such that
γτ+ (B) is bounded, then S(t) has (at least) an equilibrium point.
- 33 2.5. Minimal global B-attractors
In some applications, it happens that there exists an unbounded invariant set that
attracts all bounded sets. For this reason, we recall here the more general definition of
a minimal global attractor.
Let X be a metric space and S be a semigroup on X. Following Ladyzenskaya [La87a],
we say that a set A ⊂ X is a minimal global B-attractor if it is a nonempty, closed, set
that attracts all bounded sets of X and is minimal with respect to these properties.
The following result was noted in [HR93a].
Proposition 2.39. The semigroup S on X admits a minimal global B-attractor AX
on X if S is asymptotically smooth and if for any bounded set B ⊂ X, there exists
τ ∈ G+ such that γτ+ (B) is bounded. Moreover, AX is invariant and
[
AX = ClX ( {ω(B) | B bounded subset of X}) .
Proof. It is a direct consequence of Proposition 2.13.
If, under the assumptions of Proposition 2.39, the union of the ω-limit sets of all the
bounded sets is bounded, then AX is the compact global attractor. It was asserted in
[HR93a] that, under the hypotheses of Proposition 2.39, the minimal global B-attractor
is always locally compact. However, this is not the case as has been shown with an
example by Valero (see [ChHa]).
Consider the flow S(t) of the linear ODE ẋ = Bx where B is a n × n matrix,
which, for example, has one positive eigenvalue and n − 1 negative eigenvalues, then the
one-dimensional unstable manifold of the origin is an unbounded minimal B-attractor.
Another simple example is the flow S(t) : R2 → R2 of the ODE ẋ = 0, ẏ = −y. The
minimal global B-attractor for S(t) is the x-axis. This equation has a first integral
Φ(x, y) = x. On every level set Φc = {(x, y) | Φ(x, y) = c}, S(t) has a compact global
attractor (c, 0). This example is a special case of an evolutionary equation on a space
X, which has a continuous first integral Φ. It is often the case that on each level set
Φc , the associated semigroup S(t) admits a compact global attractor Ac . Then the
S
minimal global B-attractor AX is given by AX = ClX ( c∈R Ac ). Other examples of
such systems with first integrals are studied in [HR93b, Section 6].
Examples of minimal global B-attractors, that are not necessarily global attractors,
arise in the study of damped wave equations with local damping (see [HR93a]). An
example of an unbounded minimal global B-attractor is also given in [Qi].
- 34 Different notions of global attractors involving two different spaces are considered
in [BV89b], [MiSc], [Fe96], [Hal99] and [Mi00], for instance. Often, the semigroups S(t)
generated by evolutionary partial differential equations on unbounded domains are no
longer asymptotically smooth on function spaces, which are large enough to contain
all the interesting dynamics. One way to overcome this difficulty is to introduce two
different adequate topologies on the space X, so that, on the bounded sets for the first
topology, S(t) is asymptotically smooth for the second topology (see [Mi00], for details).
2.6. Periodic systems
In this paragraph, we very briefly indicate that the notion of global attractor can be
extended to evolutionary equations, which are nonautonomous.
Let X be a Banach space. We consider, for instance, the nonautonomous evolutionary equation
du
(t) = f (t, u) , u(s) = u0 ∈ X ,
(2.24)
dt
where f is a continuous map from R × X into X and is Lipschitz-continuous in u on the
bounded sets of X. Then, through each point (s, u0 ) of R × X, there exists a unique
local solution u(t, s, u0 , f ) of (2.24). Under appropriate hypotheses on f , this solution
is global and we set u(t, s, u0, f ) = S(t, s)u0 . The operator S(t, s) : X → X satisfies the
relations
S(s, s) = Id ,
∀s ∈ R ,
S(t, s) = S(t, τ )S(τ, s) ,
for any t ≥ τ ≥ s ,
(2.25)
and has also has the following properties
S(t, s) ∈ C 0 (X; X) ,
∀s ∈ R ,
S(t, s)u0 ∈ C 0 ([s, +∞); X) ,
∀t ≥ s ,
∀s ∈ R ,
∀u0 ∈ X .
(2.26)
For later use, we introduce a subset F ⊂ C 0 (R × X, X), which consists of functions
g(t, u) satisfying the above conditions.
If there exists ω > 0 such that f (t + ω, u) = f (t, u), for any (t, u) ∈ R × X, then
S(t + ω, s + ω) = S(t, s) ,
for any t ≥ s .
(2.27)
More generally, let (X, d) be a metric space and let us consider a family of operators
S(t, s) : X → X, s ∈ R, t ≥ s satisfying the conditions (2.25), (2.26) and (2.27). Since
- 35 S(t, s) is periodic, it is meaningful to introduce the associated period map T0 = S(ω, 0)
and study the existence of a global attractor for T0 . We begin with a lemma, which
shows that for period maps, there is an equivalent form of point dissipativness which is
easier to verify (see [Pl66] in the finite-dimesional case and [Hal00] in the general case).
Lemma 2.40. Let S(t, s) : X → X, s ∈ R, t ≥ s, satisfying the conditions (2.25),
(2.26) and (2.27). Assume that, for any bounded set B ⊂ X, there exists n0 = n0 (B)
S
such that n≥n0 T0n (∪0≤s<ω S(ω, s)B) is bounded. Then, the existence of a positive
number R such that, for any (s, u0 ) ∈ [0, +∞) × X,
lim sup kS(t, s)u0 kX ≤ R ,
(2.28)
t→+∞
is equivalent to the existence of a positive number r such that, for any (s0 , u0 ) ∈
[0, +∞) × X, there exists a time τ > s0 such that
kS(τ, s0)u0 kX < r ,
(2.29)
As a consequence of Lemma 2.40 and Theorem 2.26, we obtain the following result:
Theorem 2.41. Let S(t, s) : X → X, s ∈ R, t ≥ s, satisfying the conditions (2.25),
(2.26) and (2.27). Assume that the property (2.29) holds, that T0 is asymptotically
smooth and that, for any bounded set B ⊂ X, there exists n0 = n0 (B) such that
S
n
n≥n0 T0 (∪0≤s<ω S(ω, s)B) is bounded. Then, T0 admits a compact global attractor
A0 . Moreover, for any 0 ≤ σ < ω, Tσ = S(σ + ω, σ) has a compact global attractor
Aσ = S(σ, 0)A0 .
Remark. If S(t, s) : X → X, s ∈ R, t ≥ s, is a family of operators satisfying the
conditions (2.25), (2.26) and (2.27), we obtain a particular case of a periodic process by
setting U (τ, s)u0 = S(τ + s, s)u0 . We recall that a family of operators U (τ, s) : X → X
for τ ≥ 0, s ∈ R, is a process on X, if U (0, s) = Id, U (t, σ + s)U (σ, s) = U (σ + t, s), for
any s ∈ R and for any σ > 0, t > 0, U (t, s) ∈ C 0 (X, X) and U (t, s)u ∈ C 0 ([0, +∞), X),
for any u ∈ X and s ∈ R. The process is periodic if there exists a positive number ω
such that U (t, s + ω) = U (t, s), for any t ≥ 0, s ∈ R.
Let us remark that processes are natural extensions of the notion of continuous semigroups. Indeed, if one defines the operators Σ(t) : [s, u] ∈ R × X 7→ [s + t, U (t, s)u] ∈
R × X, Σ(t), t ≥ 0, is a continuous semigroup under the aditionnal hypothesis that
U (t, s)u is jointly continuous in (s, u).
- 36 Unfortunately, since the time variable does not belong to a compact set, the semigroup Σ(t) will never have a compact global attractor. Thus, when U (τ, s) is not
periodic, one needs to find another way to generalize the notion of compact attractor
to nonautonomous systems.
Let us come back to the evolutionary equation (2.24). For any t ≥ 0, we introduce the
translation σ(t)(f ) defined by σ(t)(f )(s, x) = f (t + s, x). We suppose now that the set
F is a metric space, with the property that f ∈ F implies that the translation σ(t)(f )
belongs to F . For any t ≥ 0, we define π(t) : X × F → X × F by
π(t)(u0 , f ) = (u(t, 0, u0, f ), σ(t)(f )) .
(2.30)
One easily shows that π(0) = Id and that π(t+s)(u0 , f ) = π(t)π(s)(u0, f ). If the family
F is chosen so that π satisfies the continuity properties required in Definition 2.1, we
have thus defined a continuous semigroup on X × F , which is called the skew-product
flow of S(t, s) or the skew-product flow of the associated process U (τ, s). Under appropriate compactness hypotheses on F , one can thus study the existence of global attractors for the continuous semigroup π(t). Skew-product flows had been first exploited
by Miller [Mill65] and Sell [Sell67] (see also [Sell71]) in the frame of ordinary differential equations. Skew-product flows have also been associated to more general processes
U (τ, s) (see [Da75]). For further study of global attractors associated to nonautonomous
systems, we refer to [ChVi1], [ChVi2], [MiSe], [Hal88], [Har91] and [SeYou].
3. General properties of global attractors
In the previous section, only a few properties of global attractors have been given. In
general, the invariance and attractivity of the global attractors, combined with some
additional hypotheses, imply interesting robustsness and regularity properties. Often
also, the flow restricted to the global attractor shows finite dimensional behaviour. In
this section, we are going to briefly describe such additional properties for compact
global attractors.
- 37 3.1. Dependence on parameters
One of the basic problems in dynamical systems is to compare the flows defined by
different semigroups. In the study of semigroups restricted to a finite dimensional
compact manifold (with or without boundary), this comparison is made very often
through the notion of topological equivalence (for the definition, see below). If the
semigroups are defined on a finite or infinite dimensional vector space for example,
then considerable care must be taken in order to discuss the behaviour of orbits at
infinity. If each of the semigroups has a compact global attractor, one can hope to
consider the topological equivalence of the flows restricted to the global attractors.
This is the strongest type of comparison of flows that can be expected in the sense
that it uses the very detailed properties of the flows. In particular, it requests the
knowledge of transversality properties, that are very difficult to show in the infinitedimensional case. For this reason, we begin with much weaker concepts of comparison,
like estimates of the Hausdorff distance between the global attractors. We shall mainly
give general comparison results and refer the reader to Section 4 and to [BV89b], [Hal88],
[Hal98], [HLR], [HR89], [HR90], [HR92b], [HR93b], [Ko90], [Ra95], [ST] and [Vi92] for
applications to (singularly) perturbed systems and discretised equations.
In this paragraph, (X, d) still denotes a metric space and we consider a family of
semigroups Sλ (t), t ∈ G+ , depending on a parameter λ ∈ Λ, where Λ = (Λ, dλ ) is a
metric space. For sake of clarity, we assume that all the semigroups Sλ are defined
on the same space X, although in many applications, each Sλ may be defined on a
different space Xλ . Then one has to determine first how to relate these spaces in order
to have a concept of convergence of the semigroups Sλ , which replaces the hypotheses
(H.1a), (H.1b) or (H.1c) given below. Such situations arise in the discretisation of
partial differential equations, in problems on thin domains etc . . . (for more details see
[HLR], [HR89],[HR93b] and [HR95]).
We assume that each semigroup Sλ has a global attractor Aλ and, if λ0 is a nonisolated point of Λ, we are interested in the behavior of Aλ when λ → λ0 .
We say that the sets Aλ are upper semicontinuous (resp. lower semicontinuous) on Λ
at λ = λ0 if
lim
λ∈Λ→λ0
δX (Aλ , Aλ0 ) = 0 ,
(resp.
lim
λ∈Λ→λ0
δX (Aλ0 , Aλ ) = 0) .
(3.1)
We say that the sets Aλ are continuous at λ0 if they are both upper and lower semicontinuous at λ0 . Due to the strong attractivity property and the invariance of the
- 38 attractors, the upper semicontinuity property holds if the dependence in the parameter
is not too bad. To get upper semicontinuity, one often assumes that either
(H.1a) There exist η > 0, τ0 ∈ G+ with τ0 > 0 and a compact set K ⊂ X such that
[
λ∈NΛ (λ0 ,η)
Aλ ⊂ K ,
(3.2)
and if λk → λ0 , xk ∈ Aλk , for k 6= 0 and xk → x0 , then Sλk (τ0 )xk → Sλ0 (τ0 )x0 ;
or
(H.1b) There exist η > 0, t0 ∈ G+ with t0 > 0 and a bounded set B0 ⊂ X such that
[
λ∈NΛ (λ0 ,η)
Aλ ⊂ B0 ,
(3.3)
and, for any ε > 0, any τ ∈ G+ , τ ≥ t0 , there exists 0 < θ = θ(ε, τ ) < η such
that
δX (Sλ (τ )xλ , Sλ0 (τ )xλ ) ≤ ε ,
∀xλ ∈ Aλ , ∀λ ∈ NΛ (λ0 , θ) .
(3.4)
In most of the cases, the hypotheses (H.1a) or (H.1b) are rather easy to check.
Actually, in the case of evolutionary partial differential equations, the compactness
condition (3.2) is often proved by showing that, due to an asymptotic smoothing effect,
the attractors Aλ are uniformly bounded with respect to λ in a metric space X2 which
is compactly embedded in the space X. Often also, the stronger convergence property
(H.1c) there exists t0 ∈ G+ with t0 > 0 such that Sλ (t)x → Sλ0 (t)x uniformly for
(t, x) in bounded sets of G+ × X2 , as λ ∈ Λ → λ0 ,
holds. Hypotheses (H.1a) or (H.1b) imply upper semicontinuity of the attractors at
λ = λ0 (see [Hal88, Theorem 2.5.2]).
Proposition 3.1. Let λ0 be a nonisolated point of Λ. If the hypothesis (H.1a) or
(H.1b) holds, the global attractors Aλ are upper semicontinuous on Λ at λ = λ0 ; that
is, limλ∈Λ→λ0 δX (Aλ , Aλ0 ) = 0.
Proof. 1) We give only the proof in the case when G+ = [0, +∞). Under the hypothesis
(H.1a), the global attractors Aλ for λ ∈ NΛ (λ0 , η) are compact. We remark that Aλ
is also the compact global attractor of the discrete semigroup Sλ (τ ). To prove upper
- 39 semicontinuity, it suffices to show that, for any sequences λk ∈ NΛ (λ0 , η), k ≥ 1,
xk ∈ Aλk , k ≥ 1, such that λk → λ0 , xk → x0 , the limit x0 ∈ Aλ0 . Since Aλk is
invariant under Sλk (τ0 ), there exists x1k ∈ Aλk such that xk = Sλk (τ0 )(x1k ). Without
loss of generality, due to the compactness of K, we can assume that the sequence x1k
converges to some element x10 . The hypothesis (H.1a) implies that x10 = Sλk (τ0 )x0 .
Using a recursion argument, one thus obtains an infinite sequence xj0 ∈ K, j → +∞,
where Sλj k (τ0 )xj0 = x0 . Clearly, the complete orbit γ(x0 ) = {Sλnk (τ0 )x0 | n ∈ Z} is
bounded in X, which implies that x0 ∈ Aλ0 .
2) Let ε > 0 be fixed. Since Aλ0 is the global attractor of Sλ0 (t), there exists a time
τε ≥ t0 such that
Sλ0 (t)B0 ⊂ NX (Aλ0 , ε/2) ,
∀t ≥ τε .
(3.5)
By hypothesis (H.1b), there exists θ > 0, such that, for λ ∈ NΛ (λ0 , θ),
δX (Sλ (τε )xλ , Sλ0 (τε )xλ ) ≤ ε/2 ,
∀xλ ∈ Aλ ,
which, together with (3.5), implies that Sλ (τε )Aλ ⊂ NX (Aλ0 , ε). Since Aλ is invariant,
Aλ ⊂ NX (Aλ0 , ε).
In general, the lower semicontinuity property does not hold, as shown by the simple
ODE
ẋ = (1 − x)(x2 − λ) ,
(3.6)
√
where λ ∈ [−1, 1]. Here, A0 = [0, 1], Aλ = 1, for λ < 0 and Aλ = [− λ, 1], for λ > 0
and a bifurcation phenomenon occurs. In general, lower semicontinuity at a given point
λ0 is obtained only by imposing additional conditions on the flow. It is mainly known to
hold in the case of gradient like systems, when all the equilibrium points are hyperbolic
(see Section 4 below). However, as pointed out in [BaPi], the lower semicontinuity
property is generic, under simple compactness assumptions. We recall that a subset
Q of a topological space Λ is residual if Q contains a countable intersection of open
dense sets in Λ. We say that a property (P) of elements of Λ is generic if the set
{λ ∈ Λ | λ satisfies (P )} is residual. Let K be a compact metric space and K c be the set
of compact subsets of K. It is well-known (see [Ku]), that, if Λ is a topological space,
and f : Λ → K c is upper semicontinuous at any λ ∈ Λ, then there exists a residual
subset Λ0 ⊂ Λ such that f is continuous at every λ0 ∈ Λ0 . Here we apply this property
to compact global attractors Aλ , satisfying the hypothesis (H.1a), with λ ∈ NΛ (λ0 , η)
replaced by λ ∈ Λ.
- 40 Corollary 3.2. If the hypothesis (H.1a) holds, with λ ∈ NΛ (λ0 , η) replaced by λ ∈ Λ,
there exists a residual subset Λ0 of Λ such that the sets Aλ are continuous at any
λ0 ∈ Λ0 .
The parameter λ can be the domain Ω ⊂ Rn on which one defines a partial differential equation. In [BaPi], Babin and Pilyugin have applied Corollary 3.2 to prove
generic continuity of global attractors of nonlinear heat equations with respect to the
domain Ω, thus recovering some of the earlier results of Henry ([He85a], [He87]).
The hypotheses (H.1a) or (H.1b) do not allow to estimate the semidistance δX (Aλ , Aλ0 ).
It becomes possible, if one imposes stronger attractivity properties on Aλ0 (see [HLR],
[BV89b, Chapter 8]):
Proposition 3.3. Assume that λ0 is a nonisolated point of Λ and that Hypothesis
(H.1b) holds. Suppose also that there exist positive constants α0 , β0 , γ0 , c0 and c1 such
that
δX (Sλ0 (t)B0 , Aλ0 ) ≤ c0 exp(−α0 t) , ∀t ≥ t0 ,
t ∈ G+ ,
(3.7)
and, for any λ ∈ NΛ (λ0 , η), for any x ∈ B0 ,
δX (Sλ (t)x, Sλ0 (t)x) ≤ c1 δΛ (λ, λ0 )γ0 exp(β0 t) , ∀t ≥ t0 ,
t ∈ G+ ,
(3.8)
then, there exist c > 0 and η1 ≤ η such that, for any λ ∈ NΛ (λ0 , η1 ), we have,
α0 γ0
δX (Aλ , Aλ0 ) ≤ cδΛ (λ, λ0 ) α0 +β0 .
(3.9)
α0 γ0
Proof. We introduce the time t1 = − ln( cc01 δ(λ, λ0 ) α0 +β0 )/α0 . We remark that t1 ≥
t0 , if η1 > 0 is small enough. From the estimates (3.7) and (3.8), we deduce that
α0 γ0
δX (Sλ (t1 )Aλ , Aλ0 ) ≤ cδΛ (λ, λ0 ) α0 +β0 , which implies (3.9) by invariance of Aλ .
The property (3.7) is difficult to verify. However, we shall prove it below for gradient
systems, whose equilibria are all hyperbolic.
Remark 3.4. If the conditions (3.7) and (3.8) hold for every λ ∈ Λ, with constants α0 ,
β0 , γ0 , c0 and c1 independent of λ, we obtain the estimate
α0 γ0
HdistX (Aλ , Aλ0 ) ≤ cδΛ (λ, λ0 ) α0 +β0 .
(3.10)
- 41 In particular, the attractors Aλ are continuous at every λ ∈ Λ.
Other concepts of comparison of the attractors Aλ , which are weaker than continuity, have been introduced in [HR93b] and may be more appropriate when the limiting
system Sλ0 is not neccessarily dissipative. For sake of clarity, we assume now that
G+ = [0, +∞); obviously the results below also hold for maps.
Definition 3.5. Let λ0 ∈ L̄, where L is a subset of Λ. The ω-limit set ω̃L (A., λ0 ) of
the family of sets Aλ , λ ∈ NΛ (λ0 , η) ∩ L, η > 0 is defined by
\
[
ω̃L (A., λ0 ) =
ClX
Aλ .
(3.11)
0<δ<η
λ∈NΛ (λ0 ,δ)∩L
Several properties of the set ω̃L (A., λ0 ) are given in [HR93b]. If λ0 ∈ L and, for each
λ ∈ NΛ (λ0 , η) ∩ L, Sλ has a compact global attractor, then the upper semicontinuity
(resp. the lower semicontinuity) of the attractors at λ = λ0 implies that ω̃L (A., λ0 ) ⊂
S
Aλ0 (resp. Aλ0 ⊂ ω̃L (A., λ0 )). If ClX ( λ∈L∩NΛ (λ0 ,η) Aλ ) is compact, it follows from
the inclusion ω̃L (A., λ0 ) ⊂ Aλ0 that the attractors are upper semicontinuous at λ0 . We
remark that the inclusion Aλ0 ⊂ ω̃L (A., λ0 ) does not imply lower semicontinuity of the
attractors Aλ . Indeed, consider the ODE ẋ = −x((−1)n λn + (x − 1)2 ) with λn = 1/n,
that is L = {1, 1/2, ..., 1/n, ...}. There is no continuity of the attractors at λ = 0;
however, ω̃L (A., 0) = A0 = [0, 1].
We notice that ω̃L (A., λ0 ) does not involve directly the semigroup Sλ0 . In particular, Sλ0
could be conservative. The following question then arises: how much information can we
obtain about a conservative system by considering the limit of dissipative systems, when
the dissipation goes to zero? We cannot hope to obtain too many specific properties of
the dynamics of the limit system in this way, but one should be able to obtain some
information about the manner in which the orbits of the dissipative systems wander
over the level sets of the energy of the limit system.
Consider the ODE u̇ = v, v̇ = f (u) − βv, where β ≥ 0 is a constant, f ∈ C 2 (R, R)
has only simple zeros and f (u) is dissipative (i.e. lim sup|u|→+∞ f (u)
u ≤ α < 0). The
Ru
2
energy functional is Φ(u, v) = (1/2)v − 0 f (s)ds. For β > 0, the ODE is a gradient
system and has a global attractor Aβ . Let {sj , j = 1, 2, ....M } be the set of the saddle
equilibrium points of the system. If Φ(sj ) 6= Φ(sk ), for j 6= k, j, k = 1, 2, ...M , then, for
any interval L = (0, β0 ],
ω̃L (A., 0) = {(u, v) ∈ R2 | Φ(u, v) ≤ cM } ,
- 42 where cM = max{Φ(sj ) j = 1, 2, ...M } (for details, see [HR93b]).
The limit ω̃L (A., λ0 ) only uses information about the attractors. As a consequence,
the transient behaviour of the semigroups Sλ for initial data not on the attractors is
completely ignored. To gain some information about this transient behaviour, one can
consider the following concept of ω-limit set:
Definition 3.6. Let λ0 ∈ L̄, where L is a subset of Λ and let Sλ (t) be a family of
semigroups on the metric space X. For a given subset B of X, the ω-limit set of B with
respect to the family of semigroups Sλ (t), λ ∈ L∩NΛ (λ0 , η), η > 0, is denoted by ω̂L (B)
and is defined in the following way: a point y ∈ ω̂L (B) if and only if there are sequences
λn ∈ L ∩ NΛ (λ0 , η), λn → λ0 , tn → +∞ and xn ∈ B such that Sλn (tn )xn → y.
One remarks that the definition of ω̂L (B) treats λ as if it were also a time parameter;
it does not prescribe an order in the limits. The notion of continuity of global attractors
or the definition of ω̃L (A., λ0 ) prescribes the limit t → +∞ before the limit λ → λ0 .
However, in many practical situations, it is not clear that an order in the limits should
be imposed (see, for instance, the discussion in [Mi99]).
Suppose that λ0 ∈ Λ and that Hypothesis (H.1c) holds. If B is a bounded set such
that the ω-limit set ωλ0 (B) of B with respect to Sλ0 exists, is nonempty, compact and
attracts B, and if, either ωλ0 (B) ⊂ B or ωλ0 (B) attracts a neighbourhood of B, then,
ωλ0 (B) = ω̂L (B). In particular, if the semigroup Sλ0 has a compact global attractor
Aλ0 ⊂ B, then the equality ω̂Λ (B) = Aλ0 holds. Applications of this property to
situations, where the limit system Sλ0 has a first integral, are given in [HR93b]. For
example, consider the retarded differential difference equation
ẋ = −(1 + ε)f (x(t)) + f (x(t − 1)) ,
(3.12)
where ε ≥ 0 is a parameter, f ∈ C 1 (R, R) satisfies f (0) = 0 and f ′ (x) ≥ δ > 0, for all
x ∈ R. For any ε ≥ 0, one defines a semigroup Sε (t) on the space X = C 0 ([−1, 0], R) by
the relation (Sε (t)ϕ)(θ) = x(t + θ, ϕ), θ ∈ [−1, 0], where x(t, ϕ) is the unique solution
of (3.12) with initial data ϕ. It is shown ([HR93b]) that, for ε > 0, the global attractor
Aε reduces to {0}, which implies that ω̃(0,ε0 ] (A., 0) = {0}, for any positive number
R0
ε0 . On the other hand, for ε = 0, the function Φ(ϕ) = ϕ(0) + −1 f (ϕ(s)) ds is a first
integral. On each level set Φ−1 (c), there is a unique equilibrium point e(c) of (3.12), for
ε = 0, and the ω-limit set ω0 (Φ−1 (c)) with respect to the semigroup S0 (t) reduces to
- 43 {e(c)}. It is then proved that, if B is an arbitrary closed, bounded set in X, we have
ω̂(0,ε0 ] (B) = ω0 (B) = I(B), where I(B) = {e(cϕ ) | ϕ ∈ B} and cϕ = Φ(ϕ).
We next want to study how the flows -restricted to the compact global attractorsvary with the perturbation parameters. If the semigroups are defined on an infinitedimensional vector space, we are lead to make severe restrictions on the flows, that we
consider. For this reason, we restrict our discussion to Morse-Smale systems. Since
general comparison results are mainly available in the frame of discrete Morse-Smale
systems, we shall restrict our study to this class.
Morse-Smale maps.
As we have already explained, in the infinite dimensional case, the strongest expected comparison of the dynamics of two different semigroups is the topological equivalence of the flows restricted to the compact global attractors. In the case of discrete
semigroups, the notion of topological equivalence is replaced by the conjugacy of the
trajectories. In [Pal69] and [PaSm], it has been proved that any Morse-Smale C r diffeomorphism S, defined on a compact manifold M is stable, that is, there exists a
neighbourhood N r (S) of S in the set Diff r (M ) of all C r -diffeomorphisms, r ≥ 1, such
that, for each T ∈ N r (S), there exists a homeomorphism h ≡ h(T ) : M → M and
h ◦ T = T ◦ h holds on M . This important stability property has been generalized to
the Morse-Smale maps defined on a Banach manifold by Oliva (see [Ol82] and [HMO]).
For sake of simplicity, we describe this result only in the case of a Banach space X. We
begin with some definitions and notations.
Let S ∈ C r (U, X), r ≥ 1 and U be an open subset of X. For any fixed point x0 of S,
we introduce the stable and unstable sets of S at x0 by
W s (x0 , S) = {y ∈ X | S n (y) → x0 as n → +∞}
W u (x0 , S) = {y ∈ X | there exists a negative orbit uy of S such that uy (0) = y
and uy (−n) → x0 as n → +∞}
(3.13)
A fixed point x0 of S is hyperbolic if the spectrum σ(DS(x0 )) does not intersect the
unit circle {z ∈ C | |z| = 1} in C with center 0.
Remark 3.7. If x0 is a hyperbolic fixed point of S ∈ C r (U, X), r ≥ 1, then there exists
- 44 a neighbourhood V of x0 in U such that the sets
s
Wloc
(x0 , S) = W s (x0 , S, V ) = {y ∈ W s (x0 , S) | S n(y) ∈ V, n ≥ 0}
u
Wloc
(x0 , S) = W u (x0 , S, V ) = {y ∈ W u (x0 , S) | S −n(y) exists and S −n (y) ∈ V, n ≥ 0}
(3.14)
are embedded C r -submanifolds of X. These sets are called the local stable and local
s
unstable manifolds of x0 . The manifold Wloc
(x0 , S) is positively invariant, whereas
u
u
Wloc
(x0 , S) is negatively invariant. Moreover Wloc
(x0 , S) is locally positively invariant.
If the part of the spectrum σ(DS(x0 )) lying outside the unit circle is composed of a
u
s
finite set of m eigenvalues, then Wloc
(x0 , S) (resp. Wloc
(x0 , S)) is an embedded C r submanifold of dimension m of X (resp. of codimension m of X).
u
If S and the derivative DS(y) are injective at each point y of ∪n≥0 (S n (Wloc
(x0 , S)),
u
n
u
r
then W (x0 , S) = ∪n≥0 (S (Wloc (x0 , S)) is an injectively immersed C -submanifold of
u
s
X, of the same dimension as Wloc
(x0 , S), and is invariant. If Wloc
(x0 , S) is of finite
codimension m, if S is injective and the derivative DS(y) has dense range at each point
s
s
(x0 , S)) is an injectively
(x0 , S)), then W s (x0 , S) = ∪n≥0 (S −n (Wloc
y of ∪n≥0 (S −n (Wloc
immersed C r -submanifold of codimension m of X (see [He81, Theorem 6.1.9]). Moreover,
W s (x0 , S) is invariant under S. For further details, see also Section 4.1.
A point x0 is a periodic point of period p if S p (x0 ) = x0 , S n (x0 ) 6= x0 , for 0 <
n ≤ p − 1. A periodic point x0 of period p is hyperbolic if the (finite) orbit O(x0 ) =
{x0 , S(x0 ), ..., S p−1(x0 )} of x0 is hyperbolic, that is, if every point y ∈ O(x0 ) is a
u
s
(y, S)
(y, S) and Wloc
hyperbolic fixed point of S p . As above, one introduces the sets Wloc
u
(y, S)), for every y ∈ O(x0 ). These stable and unstable
and W u (y, S) = ∪n≥0 (S np (Wloc
sets have the properties mentioned in Remark 3.7. Hereafter, we denote by Per(S) the
set of periodic points of S.
Let S ∈ C r (X, X), r ≥ 1. The nonwandering set Ω(S) of S is the set of all points
x ∈ J (S) (where J (S) is the maximal bounded invariant set of S) such that, given a
neighbourhood V of x in J (S) and any integer n0 , there exists n ≥ n0 with S n (V )∩V 6=
∅. If Ω(S) is finite, then Ω(S) = Per(S). One also notices that, if J (S) is compact and
S is injective on J (S), then Ω(S) is compact and invariant.
Following [Ol82] and [HMO], we introduce a topological subspace KC r (X, X) of
Cbr (X, X), r ≥ 1, with the following properties:
(KC1) S ∈ KC r (X, X) implies that J (S) is compact;
(KC2) the sets J (S) are uppersemicontinuous on KC r (X, X), that is, for any S ∈
KC r (X, X), given a neighbourhood U of J (S) in X, there exists a neighbourhood V (S)
- 45 of S in KC r (X, X) such that J (T ) ⊂ U , for any T ∈ V (S);
(KC3) for any S ∈ KC r (X, X), S and DS are injective at each point of J (S).
Example. Let Sλ ∈ Cbr (X, X), r ≥ 1, be a family of maps depending on a parameter
λ ∈ Λ, where Λ is a metric space. Assume that each map Sλ admits a compact global
attractor Aλ and satisfies the hypothesis (H.1a) or (H.1b) at every point λ0 ∈ Λ. Then,
J (Sλ ) = Aλ and the sets Aλ are uppersemicontinuous in λ. If moreover the above
condition (KC3) holds, the family Sλ , λ ∈ Λ can be chosen as a KC r (X, X)-space.
Finally, we introduce the class of Morse-Smale maps:
Definition 3.8. A map S ∈ Cbr (X, X), r ≥ 1, is a Morse-Smale map if the above
conditions (KC1) and (KC3) as well as the following conditions are satisfied:
(i) Ω(S) is finite (hence Ω(S) = Per(S));
(ii) every periodic point x0 of S is hyperbolic and dim W u (x0 , S) is finite;
s
(iii) W u (x0 , S) is tranversal to Wloc
(x1 , S), for any periodic points x0 and x1 of S.
If S is a Morse-Smale map, then J (S) = ∪x0 ∈Per(S) W u (x0 , S). The Morse-Smale
maps have a remarkable property, namely they are J -stable.
Definition 3.9. A map S ∈ KC r (X, X) is J -stable or simply stable if there exists
a neighbourhood V (S) of S in KC r (X, X), such that each T ∈ V (S) is conjugate to
S, that is, there exists a homeomorphism h = h(T ) : J (S) → J (T ) satisfying the
conjugacy condition h ◦ S = T ◦ h on J (S).
Adapting the arguments used in [Pal69] and in [PaSm], Oliva showed, mutatis
mutandis, the following basic result (see [Ol82] and [HMO]):
Theorem 3.10. Let a subspace KC r (X, X) of Cbr (X, X), r ≥ 1, be given. The set of all
r-differentiable Morse-Smale maps is open in KC r (X, X). Moreover, every Morse-Smale
map S in KC r (X, X) is J -stable.
This result has important applications in the study of partial differential equations
depending on various parameters, including time or space discretisations. In Section 4,
we shall apply it to gradient systems. If Sλ ∈ C r (X, X), r ≥ 1, is a family of maps
depending on a parameter λ ∈ Λ and Sλ0 is a Morse-Smale map, Theorem 3.10 allows
to conclude that, for λ close to λ0 , Sλ has the same type of connecting orbits. If Sλ0 is
- 46 no longer a Morse-Smale map, this persistence of connecting orbits can still be proved
in some cases, with the help of the Conley index (see [Co], [MiMr], [MR], for example).
If S1 (t) and S2 (t) are two continuous semigroups on a Banach space X, we say
that S1 (t) is topologically equivalent or simply equivalent to S2 (t), if there exists a
homeomorphism h : J (S1 ) → J (S2 ), which preserves the orbits and the sense of orientation in time t. A continuous semigroup S1 (t) is stable if there exists a neighbourhood
N (S1 (·) of S1 (·) within a given class of continuous semigroups such that every semigroup T (·) ∈ N (S1 (·)) is equivalent to S1 (·). Like above, one can define Morse-Smale
continuous semigroups. Very recently, Oliva [Ol00] has given a proof of a stability result
for Morse-Smale continuous semigroups in the infinite-dimensional case. Also, stability
of certain continuous Morse-Smale semigroups Sλ (t) generated by evolutionary equations has been proved by reducing Sλ (t) to a Morse-Smale system Σλ (t), defined by
a finite-dimensional system of ODE’s depending smoothly enough on the parameter λ
(see Section 3.4 below on inertial manifolds).
3.2. Dimension of compact global attractors
The existence of a (compact) global attractor A ⊂ X leads to the question of whether
there exists a finite-dimensional dynamical system whose dynamics on its global attractor reproduces the dynamics on A or at least whose attractor has the same topological
properties as A. Also, from the computational point of view, one is interested in knowing if the solutions on the attractor can be recovered by solving numerically a large
enough system of ODE’s and how big should be this system. A first step in this direction consists in showing that the “dimension” of the set A is finite and in giving
a good estimate of it. Various notions of dimension have been studied in conjunction
with global attractors. Among them, the Hausdorff and fractal dimensions have played
a primordial role. We will briefly describe both notions and state some results. For an
exhaustive study in the Hilbertian framework, we refer to the book of Temam ([Te]).
Let E be a topological space. We say that E has finite topological dimension if
there exists an integer n such that, for every covering U of E, there exists another
open covering U ′ refining U so that every point of E belongs to at most n + 1 sets of
U ′ . In this case, the topological dimension dim(E) is defined as the minimal integer
n satisfying this property. It is a classical result that, if E is a compact space with
dim(E) ≤ n, where n is an integer, then it is homeomorphic to a subset of R2n+1 .
Moreover, the set of such homeomorphisms is residual in the set of all maps from E
- 47 into R2n+1 . However, special properties of such embeddings are not known, and, in
the case where E is contained in a Banach space, it could be more convenient to deal
with linear projections (simply called projections in what follows). As generalizations of
the topological dimension, there are several stronger fractional measures of dimension
applicable to sets which have no regular structure. The most commonly used are the
Hausdorff and fractal dimensions.
Let E ⊂ X, where X is a metric space. The Hausdorff dimension is based on
approximating the d-dimensional volume of the set E by a covering of a finite number
of balls with radius smaller than ε, that is,
µ(E, α, ε) = inf
(
X
i
riα | ri ≤ ε and E ⊂
[
BX (xi , ri )
i
)
,
where BX (xi , ri ) is the ball of center xi and radius ri . The α-dimensional Hausdorff
measure of E is then defined as
µα (E) = lim µ(E, α, ε) ,
ε→0
and the Hausdorff dimension dimH (E) of E is essentially the value of α for which µα (E)
is a finite nonzero number,
dimH (E) = inf{α > 0 | µα (E) = 0} .
It is known [HW] that dim(E) ≤ dimH (E) and that dim(E) = dimH (E) if E is a
submanifold of a Banach space. There are also examples of sets E in Rn , for which
dim(E) = 0 and dimH (E) = n.
The fractal dimension also called limit capacity or box dimension is a stronger
measure than the Hausdorff dimension; here all the balls in the covering are required to
have the same radius. Given ε > 0, let n(ε, E) be the minimal number of balls BX (xi , ε)
of radius ε needed to cover E. One defines the fractal dimension dimF (E) as
dimF (E) = lim sup
ε→0
log n(ε, E)
.
log(1/ε)
It is easily proved that dimH (E) ≤ dimF (E) (see [Man]). The quantities dimH (E) and
dimF (E) can be different (see [Man] for an example of a compact subset K of l2 with
finite Hausdorff dimension and infinite fractal dimension).
- 48 One of the first estimates of the Hausdorff dimension of compact invariant sets
has been given by Mallet-Paret in 1976. In [MP76], he showed that, if K is a compact subset of a separable Hilbert space H and is negatively invariant under a mapping
S ∈ C 1 (U, H), where U is a given open neighbourhood of K and where, for any z ∈ K,
the derivative DS(z), restricted to some linear subspace Y ⊂ X of finite codimension,
is a strict contraction, then dimH (K) is finite. From this property, he deduced that, if
the contraction hypothesis is replaced by the condition that the derivative DS(z) is a
compact operator, for any z ∈ K, then dimH (K) is finite. The same conclusion holds if
the property on DS is replaced by the property that S(U ) is relatively compact in H.
This implies that the compact global attractor of a large class of delay equations and of
parabolic equations, including the two-dimensional Navier-Stokes equations, has finite
Hausdorff dimension (see [MP76], for an application to RFDE and the heat equation).
The hypothesis “ DS(z)Y is a strict contraction, for any z ∈ K” is a so called flattening
condition onto a finite dimensional subspace of X. Using the same ideas as Mallet-Paret
and a squeezing property of the semigroup S(t), Foias and Temam [FT79] showed that
any compact invariant subset under the flow generated by the two-dimensional NavierStokes equations has finite Hausdorff dimension and gave estimates of the dimension.
Assuming flattening conditions similar to [MP76], but on the mapping S itself, Ladyzhenskaya ([La82]) has improved the estimate of the Hausdorff dimension for the
global attractor of the two-dimensional Navier-Stokes equations given in [FT79].
Notice that, in [HMO, Theorem 6.8], there is an interesting result of existence of retarded functional differential equations the attractors of which have infinite Hausdorff
dimension.
In 1981, Mañé ([Man]) generalized the abstract result of Mallet-Paret to the case
of a Banach space and weakened the “flattening condition” in the following way.
Let X be a Banach space and Lλ (X, X) be the space of bounded linear mappings Σ that can be decomposed as Σ = Σ1 + Σ2 where Σ1 ∈ L(X, X) is compact
and kΣ2 kL(X,X) < λ. One remarks that, for Σ ∈ Lλ/2 (X, X), there exists a finitedimensional subspace Y ⊂ X such that, if ΣY : Y ⊂ X → X is the map induced by Σ
on Y , then kΣY kL(X) < λ. This amounts to define the number νλ (Σ) as the minimal
integer n such that there exists a subspace Y ⊂ X of dimension n with kΣY kL(X) < λ,
when Σ ∈ Lλ/2 (X, X).
Theorem 3.11. Let X be a Banach space, K be a compact negatively invariant set
- 49 under a mapping S ∈ C 1 (U, X), where U in an open neighbourhood of K in X. If
DS(x) belongs to the space L1 (X, X), for any x ∈ K, then
dimF (K) < +∞ .
Theorem 3.11 is proved in [Man]. There, the following explicit bound for dimF (K)
is given in the case where DS(x) belongs to the space L1/4 (X, X), for any x ∈ K,
dimF (K) ≤
log λ1
,
log(1/2(1 + ε)λ)
ν
where λ1 = ν2ν (1+ k+λ
λε ) , 0 < λ < 1/2, 0 < ε < (1/2λ)−1, k = supx∈K kDS(x)kL(X,X)
and ν = supx∈K νλ (DS(x)). One then remarks that, if DS(x) belongs to the space
L1 (X, X) for any x ∈ K, there exists an integer n, sufficiently large, such that DS n (x) ∈
L1/4 (X, X), for any x ∈ Kn ≡ ∩n0 S −j (K), which implies that dimF (K) = dimF (Kn ) is
finite.
Remark. In practice, Theorem 3.11 is very useful to show that global attractors have
finite fractal dimension, even if the bounds of the dimension given in the proof are not
so accurate. Indeed, in most of the cases where the semigroup satisfies the hypotheses of
Theorem 2.31, one can also show that the hypothesis, required on DS in Theorem 3.11,
holds.
In the case of Hilbert spaces, the notion of m-dimensional volume element and its
evolution are easily expressed, which gives much more accurate bounds of the various
dimensions of the attractor (see [DO], [CFT85], where DS(x) is a compact mapping
and [GT87b] for the non compact case). In particular, the Lyapunov exponents of the
flow on the attractor have become a standard tool in the description of the evolution of
volumes under the semigroup S. A result, first proved by Constantin and Foias [CF85]
in the case of the attractor of the two-dimensional Navier-Stokes equations, states that,
if the sum of the first m global Lyapunov exponents on an invariant compact set K is
negative, then the Hausdorff dimension of K is less than m and the fractal dimension
of K is finite and bounded above, up to the product by a universal constant, by m
(see [CFT85, Chapter 3] and, for refinements using the local Lyapunov exponents,
[EFT]). We refer especially the reader to the book of Temam [Te], where these topics
are well explained and estimates of the dimension of the global attractors of numerous
partial differential equations, including the reaction-diffusion equations, the damped
- 50 wave equations, the Navier-Stokes, Kuramoto-Sivashinsky and Cahn-Hilliard equations,
are given in terms of physical parameters. Other various estimates of dimensions of
attractors are contained in [BV89b], [La87b], [La90] (see also [BN] in this volume).
Finally, we note that Thieullen [Th] has given estimates of the Hausdorff dimension of
K, involving “Lyapunov exponents” in the frame of Banach spaces.
If K is a compact subset of a Banach space X, with dimF (K) < m/2, m ∈ N, then,
for every subspace Y ⊂ X of dimension dim Y ≥ m, the set of projections P : X → Y
such that P Y is injective on Y is a residual subset of the space of all the (continuous)
projections from X onto Y , endowed with the norm topology. This result has been given
in [Man], where, by an unfortunate mistake, dimF (K) has been replaced by dimH (K).
One notices that the statement is no longer true with the hypothesis dimH (K) < m/2
(see [SYC]). Recently, in the Hilbertian case, the above result has been improved by
Foias and Olson [FO], who showed that the inverse (P Y )−1 of most projections P
are Hölder continuous mappings. In general, these inverse are not Lipschitz-continuous
([ML]).
Theorem 3.12. (Mañé; Foias and Olson). Let X be a Hilbert space and K be a
compact subset of X with fractal dimension dimF (K) < m/2, m ∈ N. Then, for any
projection (resp. orthogonal projection) P0 onto a subspace Y of X of dimension m
and for any ε > 0, there exist a projection (resp. orthogonal projection) P onto Y and
a positive number θ ≤ 1, such that P K is injective, kP − P0 kL(X,Y ) ≤ ε and P K has
Hölder inverse, i.e.,
kP −1 x − P −1 ykL(X,X) ≤ Ckx − ykθX ,
x, y ∈ P (K) .
As an application of this theorem, we go back to Example 2.2, in the Hilbertian
case, and write the equation (2.1) under the following short form, where we assume, for
sake of simplicity, that Y = X,
du
= F (u) ,
dt
u(0) = u0 ∈ X .
(3.15)
We assume that the continuous semigroup S(t) has a compact global attractor A of
finite fractal dimension and that F A is Hölder continuous from X into X. Let P be a
projection given by the above theorem. On P A, the following dynamical system is well
defined,
dz
= P F (P −1 z) ≡ F̃ (z) , z ∈ P A ,
(3.16)
dt
- 51 where, under the above hypothesis, F̃ is Hölder continuous from P A into P A. The next
step is to extend this system to a system of differential equations defined everywhere in
Rm , by using a standard extension theorem like the theorem of Stein. Since F̃ is only
Hölder continuous, the solutions of this generalized system of ODE’s may not be unique
and differentiable. However, one can show that the solutions of this extended system
exist globally and are attracted by P A (see [EFNT, Chapter 10]).
One can also use Theorem 3.12 to show that, under the above hypotheses, it is always
possible to reproduce “approximately” the dynamics of (3.15) on the global attractor
A by a system of ODE’s in R3 . This is done in [Ro], where it is proved that, if F is
bounded and continuous on A, then, for any T > 0 and any ε > 0, there exist two
functions g : R3 → R3 and Φ : R3 → X, which are Lipschitz and Hölder continuous,
respectively, such that, for any solution u(t) ∈ A, there exists a solution z(t) of
dz
= g(z) ,
dt
with kΦ(z(t)) − u(t)kX ≤ ε, for any t ∈ [0, T ].
If one drops the requirement of obtaining a conjugacy between the flows on the
attractors, one can construct a homeomorphism between the attractor A and the one
of a finite sytem of ODE’s. More precisely, if A is the compact global attractor of
a continuous semigroup generated by an evolutionary equation on a Hilbert space X,
with finite Hausdorff dimension, then there exists a finite system of ODE’s with a
global attractor K, homeomorphic to A. Conversely, if K is the global attractor of
a finite system of ODE’s, there exists an infinite-dimensional continuous semigroup
S(t) generated by an evolutionary equation on X, with a compact global attractor A
homeomorphic to K and having finite Hausdorff dimension (see [Ro], where also a review
of other related results is given).
3.3. Regularity of the flow on the attractor and determining modes
If S(t) is a continuous semigroup on X, which has a compact global attractor A, the
semigroup S(t) restricted to A can have interesting “regularity” properties that are not
shared by the semigroup S(t) and are a consequence of the compactness and invariance
of A. For instance, if S(t) is generated by a partial differential equation defined on
a domain Ω ⊂ Rn , A can be composed of functions u(y) which are more regular in
the space variable y ∈ Ω than the typical elements of the space X. On the other
hand, for semigroups defined by evolutionary equations in the infinite-dimensional case,
- 52 it is not expected, in general, that, for each x ∈ X, S(t)x is differentiable in t, for
t ≥ 0. In many cases, however, the function t 7→ S(t)x will be differentiable in t if x
belongs to a compact invariant set. Time regularity results in a general setting had
been obtained in 1985 by Hale and Scheurle ([HS]) and have recently been generalized
in [HR00], through the use of a Galerkin decomposition. We present here these results
in the simplified situation of Example 2.2. In the same time, we want to emphasize the
relation between regularity and finite-dimensionality character. We recall that X is a
Banach space, A is the infinitesimal generator of a C 0 -semigroup Σ0 (t) in X, and that
f : Y → X is a Lipschitzian mapping on the bounded sets of Y , where either
1) Y = X, or
2) exp(At) is an analytic linear semigroup on X and Y = X α = D((λId − A)α ), with
α ∈ [0, 1), λ an appropriate real number.
In the case 1), we set Y = X α for α = 0. Assuming that all of the solutions of the
following equation on Y ,
du
(t) = Au(t) + f (u(t)) ,
dt
t>0,
u(0) = u0 ∈ Y ,
(3.17)
are global, we obtain a continuous semigroup S(t) on Y .
If Σ0 (t) is an analytic semigroup, time regularity properties of the mapping S(t)x,
for x ∈ Y , are well-known (see [He81, Chapter 3]). Actually, if f : u ∈ Y 7→ f (u) ∈ X
is of class C k or analytic, then, for t > 0, the mapping t → S(t)x, x ∈ Y , is of class
C k or analytic. Time regularity properties have been especially addressed in the case
of the Navier-Stokes equations and other parabolic systems (see [FT79], [FT89], [Pr91],
for example). For this reason, in the next theorem, we consider only the case where
Y = X. We suppose that there exists a positive number θ such that the radius of the
essential spectrum r(σess (exp At)) satisfies
r(σess (exp At)) ≤ exp(−θt) ,
∀t ≥ 0 .
(3.18)
The following regularity result is given in [Hal88] and follows from results proved in
[HS].
Theorem 3.13. Suppose that A satisfies the condition (3.18), f ∈ C k (X, X), 1 ≤ k <
+∞ (resp. f is analytic from X to X) and that S(t) has a compact global attractor A.
Then, there exists a positive number η depending on A such that, if
kDf (u)kL(X,X) ≤ η , for any u in a neighbourhood of A ,
(3.19)
- 53 the mapping t : R 7→ S(t)u ∈ X is of class C k (resp. analytic), for any u ∈ A.
The proof of Theorem 3.13 is rather long. The most important steps of the proof
will be described below, when we indicate how the above theorem can be generalized.
The above time regularity property in Theorem 3.13 cannot be true if f is not at least
a C 1 -function, as illustrated by the contre example given in [Hal88, page 57].
The smallness condition (3.19) is rather restrictive and limits the applications of this
result. However, if one exploits further the ideas contained in [HS] and uses a Galerkin
decomposition, one can generalize Theorem 3.13 in a significant way. For example, if X
is a Hilbert space and the eigenfunctions of the operator A form a complete orthonormal
system, we can decompose any solution u of (3.17) into the sum of two functions vn
and wn , with vn being the sum of the first n terms in the expansion of u and wn being
the remainder. Under additional natural conditions on A and f , one shows that there
exists an integer N1 such that each solution on the compact global attractor can be
represented in the form
u(t) = vN1 (t) + wN1 (t) = vN1 (t) + w∗ (vN1 )(t) ,
(3.20)
where the function w∗ (vN1 )(t) depends on vN1 (s), s ≤ t and is as smooth in vN1 as
the vector field f . The equation (3.17) is thus reduced to a finite-dimensional system
of N1 (nonautonomous) functional differential equations with infinite delay. It is very
natural to refer to the coefficients of the projection vN1 (t) of u(t) onto the first N1
eigenfunctions as the determining modes of the equation (3.17) on the attractor A.
Furthermore, wN1 (t) = w∗ (vN1 )(t) is a solution of an equation similar to (3.17), where,
if N1 is large enough, the nonlinearity satisfies the smallness condition (3.19). This
property, together with the reduction of (3.17) to a finite-dimensional system of FDE’s,
will imply the regularity in time of the solutions on the attractor A.
We now state these results more precisely. We keep the assumptions of Example 2.2,
which have been recalled above and assume further that
H1 f : Y → X is a C k function, k ≥ 1;
H2 the continuous semigroup S(t) has a compact invariant set J ;
H3 For any n ≥ 1, n ∈ N, there is a continuous linear map Pn : Z → Z, where Z = X
or Y , such that APn = Pn A on D(A), and the following additional properties hold:
(i) Pn converges strongly to the identity in Z as n goes to infinity;
(ii) there exists a positive constant K0 ≥ 1 such that, if Qn = Id − Pn , then,
kPn kL(Z,Z) ≤ K0 ,
kQn kL(Z,Z) ≤ K0 ,
∀n ∈ N ;
(3.21)
- 54 (iii) there exist an integer n1 , two positive constants δ1 and K1 ≥ 1 such that, for
n ≥ n1 , t > 0, we have,
keAt wkZ ≤ K1 e−δ1 t kwkZ , ∀w ∈ WnZ ,
keAt wkY ≤ K1 e−δ1 t t−α kwkX , ∀w ∈ WnX ,
(3.22)
where WnZ = Qn Z, for Z = X or Y . We also set VnZ = Pn Z.
In addition, our main result will depend upon either,
H4 there are a positive constant K2 ≥ 1, independent of n ∈ N, and a sequence of
positive numbers δn , δn →n→+∞ +∞, such that, for t > 0,
keAt wkZ ≤ K2 e−δn t kwkZ , ∀w ∈ WnZ ,
keAt wkY ≤ K2 e−δn t (t−α + δnα )kwkX , ∀w ∈ WnX ,
(3.23)
or
H5 the set {Df (u1 )u2 | u1 ∈ J , ku2 kY ≤ 1} is relatively compact in X.
Remark 3.14. In several concrete situations, there exists a neighbourhood U of J in
Y such that f : U → X is completely continuous. If X is a reflexive Banach space, one
shows, by arguing as in [MP76, page 339] that this property implies the hypothesis H5.
We recall that, for any k > 0, BZ (0, k) denotes the open ball in Z of center 0
and radius k. Let r be a positive constant such that kukY ≤ r/K0 , for any u ∈ J .
k
In what follows, we denote by Cbu
(BY (0, 4r); X) the subset of C k (BY (0, 4r); X) of the
mappings g whose derivatives Dj g(u), j ≤ k, are bounded for u ∈ BY (0, 4r) and the
derivative Dk g(u) : u ∈ BY (0, 4r) 7→ Dk g(u) ∈ Lk (Y, X) is uniformly continuous. To
prove that the restriction of the flow S(t) to J is of class C k , k ≥ 1, we shall assume
k
(BY (0, 4r); X).
that, f ∈ C k (Y, X) belongs to the space Cbu
If u(t) is a mild solution of (3.17) contained in J and
u(t) = Pn u(t) + Qn u(t) ≡ v(t) + w(t) ,
(3.24)
then (v(t), w(t)) is a mild solution of the following system
dv
= Av + Pn f (v + w) ,
dt
dw
= Aw + Qn f (v + w) .
dt
(3.25)
- 55 If n ≥ n1 , the property (3.22) implies that
Z t
eA(t−s) Qn f (v(s) + w(s)) ds .
w(t) =
(3.26)
−∞
In what follows, we shall always choose n ≥ n1 . For d > 0 and n ≥ 1, we introduce the
following sets,
UPn Y (J , d) = {v ∈ VnY | kvkY < 2r , distY (v, Pn J ) < d} ,
UQn Y (d) = {w ∈ WnY | kwkY < inf(d, 2r)} .
as well as the subsets
CP0 n Y (J , d) = C 0 (R; UPn Y (J , d)) ,
0
CQ
(d) = C 0 (R; UQn Y (d)) ,
nY
k
(R; UPn Y (J , d)) ,
(J , d) = Cbu
CPk,u
nY
k,u
k
CQ
(d) = Cbu
(R; UQn Y (d)) ,
nY
where k ≥ 1.
The equality (3.26) suggests that, given v ∈ CP0 n Y (J , d), the function w(t) can be
0
0
(d) defined by
(d) → CQ
obtained as a fixed point of the operator Tv (w) : CQ
nY
nY
Tv (w) =
Z
t
eA(t−s) Qn f (v(s) + w(s)) ds .
(3.27)
−∞
The problem consists now in finding d > 0 and N0 ≥ n1 , such that, under the above
0
(d) into itself,
hypotheses, the mapping Tv is indeed a uniform contraction from CQ
nY
for n ≥ N0 . Actually, motivated by the next remark, we shall also prove that Tv is a
0,u
(d) into itself, for n ≥ N0 .
uniform contraction from CQ
nY
Remark. If J is a compact invariant set for S(t), the set of the complete orbits u(t)
of (3.17) contained in J is uniformly equicontinuous; that is, for any positive number
η0 , there exists a positive number η1 such that, for any t ∈ R, for any complete orbit
u(R) ⊂ J , ku(t + τ ) − u(t)kY ≤ η0 if |τ | ≤ η1 .
If the hypotheses H4 or H5 hold, one can prove that, for n large and d small
0
enough, the map Tv is a uniform contraction from CQ
(d) into itself [HR00].
nY
Theorem 3.15. Assume that the hypotheses H1, H2, H3 and either H4 or H5 hold.
Then, there is a positive constant d1 such that, for 0 < d ≤ d1 , there exist an integer
N0 (d), and, for n ≥ N0 (d), a (unique) Lipschitz-continuous function
0
w∗ : v ∈ CP0 n Y (J , d) 7→ w∗ (v) ∈ CQ
(d) ,
nY
- 56 which is a mild solution of
dw∗ (v)
(t) = Aw∗ (v)(t) + Qn f (v(t) + w∗ (v)(t)) .
dt
(3.28)
The mapping w∗ (v)(t) depends only upon v(s), s ≤ t, and the map w∗ is also Lipschitz0,u
(d).
(J , d) into CQ
continuous from CP0,u
nY
nY
∗
k
Moreover, w (v) is as smooth in v as the vector field f ; that is, if f ∈ Cbu
(BY (0, 4r); X),
0,u
0,u
∗
k
k ≥ 1, then the mapping w is in Cbu (CPn Y (J , d); CQn Y (d)).
k
Furthermore, if f ∈ Cbu
(BY (0, 4r); X), k ≥ 1, then w∗ is a uniformly continuous map
k,u
from CPk,u
(J , d) into CQ
(d).
nY
nY
Remark 3.16. Under the hypotheses of Theorem 3.15, we can choose N1 = N1 (d1 ) ≥
N0 (d1 ), such that, if u(t) ∈ J , for t ∈ R, is a mild solution of (3.17), then, for n ≥ N1 ,
0
w(t) = Qn u(t) belongs to CQ
(d1 ) and thus, by uniqueness of the mild solutions, u(t)
nY
must be represented as
u(t) = v(t) + w∗ (v)(t) ,
(3.29)
where v(t) = Pn u(t) is the mild solution of the system of functional differential equations
dv
(t) = Av(t) + Pn f (v(t) + w∗ (v)(t)) .
dt
(3.30)
0,u
(d1 ) respectively.
(J , d1 ) and CQ
Moreover, v(t) and w(t) are in CP0,u
nY
nY
If, for any n, the range of Pn is of dimension n, the above property leads to say that
the flow on the compact invariant set J is determined by a finite number N1 of modes.
More classically, one says that the equation (3.17) has a finite number N1 of determining modes if the range of PN1 is finite-dimensional of dimension N1 and if the
condition
kPN1 u1 (t) − PN1 u2 (t)kY →t→+∞ 0 implies that ku1 (t) − u2 (t)kY →t→+∞ 0 .
The property of finite number of determining modes has been extensively studied for
parabolic equations and, especially for equations arising in fluid dynamics, like the
Navier-Stokes equations. We refer to Foias and Prodi [FP67] and to Ladyzhenskaya
[La72] for the earliest results on the two-dimensional Navier-Stokes equations. Later,
various estimates for the minimal number of determining modes in terms of the Reynolds
number or the Grashof number have been established (see the estimates in [FMTeTr],
- 57 which have been improved in [JTi93]). A generalisation of the concept of determining
modes to the one of determining functionals together with applications to dissipative
parabolic equations has been recently given by Cockburn, Jones and Titi [CJTi97].
Based on the ideas of [CJTi97], Chueshov [CHu98] extended their results to other dissipative equations, including the damped wave equation.
Here the property of finite number of determining modes is a direct corollary of
Theorem 3.15, if the compact invariant set J is moreover the global attractor ([HR00]).
In the next theorem, we do not need the assumption that PN1 has finite-dimensional
range. Of course, in view of the applications, the case when PN1 has finite-dimensional
range is more interesting.
Theorem 3.17. Assume that the compact invariant set J is the global attractor of
the equation (3.17) and that the hypotheses H1, H2, H3 and either H4 or H5 hold.
If u1 (t) and u2 (t) are any two solutions of (3.17) satisfying
kPN1 u1 (t) − PN1 u2 (t)kY →t→+∞ 0 ,
(3.31)
where the integer N1 has been defined in Remark 3.16, then
ku1 (t) − u2 (t)kY →t→+∞ 0 .
(3.32)
We now go back to the regularity in time of the complete orbits contained in J .
Arguing as in [HS, page 154] by considering the auxiliary differential equation
dv
= PN1 Av + PN1 f (v + w∗ (v)) ,
ds
v(0) = v0 ,
0
with v0 given in the Banach space Cbu
(R; PN1 Y ) and using Theorem 3.15 together with
Remark 3.16, we obtain [HR00]:
Theorem 3.18. Assume that the hypotheses H1, H2, H3 and either H4 or H5
hold and that, for any n ≥ 1, APn is a linear bounded mapping from Y into Y . If
k
f belongs to Cbu
(BY (0, 4r); X), k ≥ 1, then, for any u0 ∈ J of (3.17), the mapping
k
t ∈ R 7→ S(t)u0 = u0 (t) ∈ Y belongs to Cbu
(R; Y ) and u0 (t) is a classical solution of
(3.17). Moreover, there exists a positive constant Kk,J such that, for any u0 ∈ J ,
sup k
t∈R
dj u0
(t)kY ≤ Kk,J ,
dtj
∀j , 1 ≤ j ≤ k .
(3.33)
- 58 To show that the restriction of the flow S(t) to J is analytic when f is analytic,
we complexify the spaces X and Y , the operators A, Pn , Qn etc . . . and assume, as in
[HS], that
H6 there exists a real number ρ > 0 such that f has an holomorphic extension
from DY (4r, ρ) = {u1 + iu2 | u1 ∈ BY (0, 4r) , u2 ∈ BY (0, ρ)} into the complexified
space X and f is bounded on the bounded sets of DY (4r, ρ).
We also complexy the time variable. Given a small positive number θ, we introduce
the complex strip Dθ = {t ∈ C | |Im t| < θ} and the Banach space Cθ (Z), defined by
Cθ (Z) = {u : Dθ → Z | u is continuous, bounded in Dθ , holomorphic in Dθ ,
and u(t) ∈ R , ∀t ∈ R} ,
and equipped with the norm k|uk|Z = supt∈Dθ ku(t)kZ .
Showing first an analytic analog of Theorem 3.15 and arguing as in the proof of
Theorem 3.18 yield:
Theorem 3.19. Assume that the hypotheses H1, H2, H3, H6 and either H4 or H5
hold and that, for any n ≥ 1, APn is a linear bounded mapping from Y into Y . Then,
there exists θ > 0, such that any solution u0 (t) ⊂ J of (3.17) belongs to Cθ (Y ). In
particular, t ∈ R 7→ u0 (t) ∈ Y is an analytic function.
Theorem 3.18 and Theorem 3.19 can be applied to several evolutionary PDE’s,
including the heat and damped wave equations, and even to PDE’s with delay. In these
cases, regularity in time (up to analyticity) on the compat attractor is obtained, even
if the solutions are not regular in the spatial variable, that is, even if the domain Ω, on
which the equation is given, is not regular (see Section 4.5 below, for the time regularity
of the flow on the global attractor of the linearly damped wave equation).
In [HR00], Theorem 3.15, Theorem 3.18 and Theorem 3.19 are proved under more general conditions than H4 or H5, which allows applications to weakly dissipative equations
like the weakly damped Schrödinger equation. Also there the commutation condition
APn = Pn A is relaxed and application to the wave equation with local damping is given.
When the evolutionary equation (3.17) arises from a partial differential equation
defined on a domain Ω ∈ Rn , one deduces regularity in the spatial variable from Theorem 3.15 and Theorem 3.18. Indeed, under the hypotheses of Theorem 3.18, one shows
j
that, for 1 ≤ j ≤ k − 1, for any solution u0 (t) ∈ J of (3.17), the derivative ddtuj0 belongs
- 59 to Cb0 (R; D(A)) and is a classical solution of the equation
dj+1 u
dj u
dj f (u)
(t)
=
A
(t)
+
(t) ,
dtj+1
dtj
dtj
t>0,
dj u
(0) given in Y ,
dtj
(3.34)
In the general case, due to the boundary conditions, u ∈ J is not expected to belong
to D(Aj ), for j ≥ 2. However, we can obtain higher order regularity results ([HR00]),
as it is illustrated in the next example where A is the generator of a C 0 -semigroup only,
Y = X and α = 0.
We introduce a family of spaces Zl , l ∈ N, with Zl+1 ⊂ Zl , Z0 = X, D(A) ⊂ Z1 , such
that,
Au = g ,
u ∈ D(A) ,
g ∈ Zl ,
implies
u ∈ Zl+1 .
(3.35)
A simple recursion argument using Theorem 3.18 shows the next regularity result.
Theorem 3.20. Assume that Y = X, α = 0, that (3.35) as well as the hypotheses H1,
H2, H3 and either H4 or H5 hold and that, for any n ≥ 1, APn is a linear bounded
k
mapping from X into X. Suppose that f belongs to Cbu
(BX (0, 4r); X), k ≥ 1. If, for
k−j−1
k ≥ 2, f is in Cb
(Zj ; Zj ), for 1 ≤ j ≤ k − 1, then, for any orbit u0 (t) ⊂ J , the
mapping t ∈ R 7→ u0 (t) belongs to Cbj (R; Zk−j ), 0 ≤ j ≤ k. And there exists a positive
constant K̃Jk such that, for any u0 (t) ⊂ J ,
ku0 kCj (R;Zk−j ) ≤ K̃Jk ,
b
∀j ,
0≤j≤k .
(3.36)
We remark that regularity in Gevrey classes can also be deduced from Theorem 3.15, Theorem 3.18 and Theorem 3.20 (for earlier regularity results in Gevrey
classes in the case of dissipative equations, we refer to [FT89], [Pr91], [FeTi98], [OTi],
for example).
Finally, it should be noticed that the proofs of time regularity in [FT79], [FT89], [Pr91]
use a classical Galerkin procedure. Also, in his proof of the space regularity of the
global attractor of the weakly damped Schrödinger equation, Goubet has introduced a
Galerkin decomposition [Go96], [Go98], in a spirit different from the above one.
- 60 3.4. Inertial manifolds
Theorem 3.12 and Theorem 3.15 of the previous sections allow to embed the compact
global attractor of some classes of systems into the one of a finite-dimensional system of
differential equations. In the second case, we obtained functional differential equations
with infinite delay, while in the first case, we got ordinary differential equations, the
solutions of which may not be unique. It is therefore natural to try to exhibit classes of
dissipative systems, for which these ODE’s define a flow. This is actually the purpose
of the theory of inertial manifolds, introduced by Foias, Sell and Temam [FST].
Suppose that we are given a continuous semigroup S(t) on a Banach space X,
which is generated by an evolutionary equation on X and has a compact global attractor
A. One can define an inertial manifold M of S(t) as a finite-dimensional Lipschitzian
submanifold of X, which contains A and is positively invariant under S(t) (i.e. S(t)M ⊂
M, for any t ≥ 0). If the semigroup S(t) is one-to-one on M, then the flow restricted
to M is determined by a finite-dimensional system of ODE’s with locally Lipschitzian
vector field. This finite-dimensional system is called inertial form. As indicated below,
in the process of constructing inertial manifolds, one shows that M attracts bounded
sets exponentially. For this reason probably, Foias, Sell and Temam have included the
exponential attraction property of bounded sets, in their definition.
Up to the present time, one of the basic ways to construct inertial manifolds has
been to obtain the inertial manifold as a Lipschitzian graph over a finite-dimensional
space and to apply the classical methods of center manifold theory. With such methods,
the construction of inertial manifolds encounters the same technical problems and the
same obstructions as the one of global center manifolds. In order to prove the existence
of such global center manifolds, one needs some normal hyperbolicity property of the
manifold M, that is, the flow in X towards M must be stronger than the dispersion of
the flow on M. Since M does not only contain the neighbourhood of equilibrium points
(as it is the case for local center manifolds of equillibria) and that A may be large, the
dispersion of the flow on M may be large. In the frame of semigroups generated by
partial differential equations, this strong normal hyperbolicity requirement leads to the
so-called cone condition ([MPS]) and gap condition (see [FST], [CL88], for example),
that we explain below. Unfortunately, these conditions are shown to be satisfied only
by some partial differential equations in one space variable, including the reactiondiffusion equations (see [FST], [Te]), the Ginzburg-Landau equation [Te], the CahnHilliard equation, the Kuramoto–Sivashinsky equation, etc · · · and by few reaction-
- 61 diffusion equations on special domains in dimensions 2 and 3 ([MPS], [HR92c]).
To be more specific, we go back to the evolutionary equation of Example 2.2 with
Y = X α , α ∈ [0, 1). We assume that the spectrum of A consists of a sequence of real
eigenvalues λn , n ∈ N, in increasing order. For any integer n, we introduce the spectral
projection Pn ∈ L(X, X) onto the space generated by the eigenfunctions associated with
the first n eigenvalues and assume that the hypotheses H1, H2, H3 and H4 of Section
3.3 hold with δn = λn+1 and J = A. In particular, these assumptions hold if A is a
positive self-adjoint operator with compact inverse.
In order to obtain a globally Lipschitzian nonlinear function on the right hand side
of (2.1), we truncate the function f . Let m : R → [0, 1] be a C ∞ -function such that
m(y) = 1 if y ∈ [0, 1], m(y) = 0 if y ≥ 2. Let r > 0 be chosen so that A ⊂ B Y (0, r). We
then set
kuk2Y
fm (u) = m
f (u) ,
4r 2
and consider the modified equation
du(t)
= Au(t) + fm (u(t)) ,
dt
t>0,
u(0) = u0 ∈ Y .
(3.37)
Clearly, the function fm is globally Lipschitzian and bounded from Y into X and (3.37)
also defines a continuous semigroup on Y , denoted by Sm (t). One may construct an
inertial manifold for Sm (t) by using a Galerkin method. Like in Section 3.3, we choose
an integer n and write any solution u(t) of (3.37) as u(t) = Pn u(t)+Qn u(t) = v(t)+w(t),
where (v, w) is a solution of the system
dv
= Av + Pn fm (v + w) ,
dt
dw
= Aw + Qn fm (v + w) .
dt
(3.38)
One way for obtaining an inertial manifold M of Sm (t) as a graph over VnY is to solve,
for every v0 ∈ VnY the system (3.38) on (−∞, 0], under the condition
v(0) = v0 ,
w ∈ Cb0 ((−∞, 0), WnY ) .
(3.39)
Due to (3.26), given v0 ∈ VnY , (v(t), w(t)) is a solution of (3.38) and (3.39) if and only
if w(t) is a fixed point of the map Tv0 : Cb0 ((−∞, 0), WnY ) → Cb0 ((−∞, 0), WnY ) given by
Tv0 (w) =
Z
t
−∞
eA(t−s) Qn fm (v(s) + w(s)) ds ,
- 62 where v(t) is the solution of
dv
= Av + Pn fm (v + w) ,
dt
v(0) = v0 .
(3.40)
If we prove that Tv0 : Cb0 ((−∞, 0), WnY ) → Cb0 ((−∞, 0), WnY ) is a strict contraction,
when Cb0 ((−∞, 0), WnY ) is equipped with the norm kwkµ = supt≤0 (eµt kw(t)kY ), where
µ > 0 is well chosen, then Tv0 has a unique fixed point wv0 (t). One then checks that
the graph of the Lipschitzian mapping Ψ : v0 ∈ VnY 7→ Ψ(v0 ) = v0 + wv0 (0) ∈ Y defines
an inertial manifold M of Sm (t). For example, Tv0 is a strict contraction, if
λn+1 − λn ≥ Cλα
n+1 ,
(3.41)
where C is a positive constant depending on the Lipschitz constant of fm and on α.
The condition (3.41) is a gap condition on the eigenvalues of A and is rather restrictive.
Due to the positive invariance of the inertial manifold M, the equation (3.37) on
M reduces to the finite system of ODE’s:
dv
(t) = Av(t) + fm (Ψ(v(t))) ≡ g(v(t)) ,
dt
t>0,
v(0) given in VnY .
(3.42)
This system defines a flow on VnY = Pn Y and has a compact global attractor An = Pn A.
The solutions of (3.42) on Pn A are written as v(t) = Pn Sm (t)Ψ(v(0)), that is, the flow
of (3.42) on Pn A is conjugate to Sm (t).
Suppose now that S(t) = Sλ (t) and also Sm (t) depend on a parameter λ in a
Banach space Λ and that, for each λ, one can construct an inertial manifold Mλ over
VnY and an inertial form
dv
= gλ (v(t)) ,
dt
t>0,
v(0) given in VnY ,
where gλ and Dgλ are continuous functions of v, λ. If the flow defined by the vector
field gλ0 is structurally stable, then we know that each flow Sλ (t)A is equivalent to
λ
the flow of Sλ0 (t)A , for λ close to λ0 (see [HR92c] and [MSM]).
λ0
- 63 -
4. Gradient systems
Until now, we have not described the behaviour of the flow on the global attractor.
Even in the finite-dimensional case, this behaviour is often not known. In the case of
the gradient systems, a partial description of the flow restricted to the attractor can be
given. We first recall some general properties of gradient systems and then present a
few examples of such systems.
4.1. General properties of gradient systems
We recall that the set G+ denotes either [0, +∞) or N
Definition 4.1. Let S(t), t ∈ G+ be a semigroup on X.
1) A function Φ ∈ C 0 (X, R) is a Lyapunov functional if
Φ(S(t)u) ≤ Φ(u) ,
∀t ∈ G+ ,
∀u ∈ X .
(4.1)
2) A Lyapunov functional Φ is a strict Lyapunov functional if, moreover,
Φ(S(t)u) = Φ(u) , ∀t ∈ G+ , implies that u is an equilibrium point .
(4.2)
3) A semigroup S(t) is a gradient system if it has a strict Lyapunov functional and if,
either G+ = N or G+ = [0, +∞) and S(t) is a continuous semigroup. In the later case,
S(t) is called a continuous gradient system.
The simplest example of discrete gradient system is a monotone map S on R (for
example, Sx ≤ Sy if x ≤ y).
Notation. We denote E = {z ∈ X | S(t)z = z , ∀t ≥ 0} the set of equilibrium points of
S(t). Clearly, E is an invariant and closed set. If S is a discrete semigroup, E is simply
the set of fixed points Fix(S) of S.
The following result, known as the Invariance Principle of LaSalle, plays a basic
role in the theory of gradient systems. We set G− = {−g | g ∈ G+ }.
Proposition 4.2. Invariance Principle of LaSalle
Let S(t) be a gradient system on X with a Lyapunov functional Φ.
1) If z is an element of X such that γτ+ (z) is relatively compact in X, then,
i) l = limt→+∞ Φ(S(t)z) exists and Φ(v) = l, for any v ∈ ω(z),
- 64 ii) ω(z) ⊂ E; in particular, E =
6 ∅, and δX (S(t)z, E) →t→+∞ 0.
0
−
2) Let uz ∈ C (G , X) be a negative orbit through some z ∈ X. If the negative orbit
uz (G− ) is relatively compact, then the set αuz (z) is non-empty, compact, invariant,
consists only of equilibrium points and δX (uz (−t), αuz (z)) →t→+∞ 0. Furthermore,
αuz (z) is connected.
Proof. The function t 7→ Φ(S(t)z) is non increasing and bounded from below, since
Φ(.) is a continuous function on X and that γτ+ (z) is relatively compact in X, hence
l ≡ limt→+∞ Φ(S(t)z) exists. If v ∈ ω(z), there exists a sequence tn → +∞ such that
S(tn )z → v. As Φ(.) is continuous on X, Φ(S(tn )z) → Φ(v) and Φ(v) = l.
One remarks that the Property i) also holds if the Lyapunov functional Φ is not a strict
Lyapunov functional.
Since γτ+ (z) is relatively compact, the semigroup S(t) restricted to γτ+ (z) is asymptotically smooth. From Proposition 2.13, we then deduce that ω(z) 6= ∅ and that
δX (S(t)z, ω(z)) →t→+∞ 0. The inclusion S(t)ω(z) ⊂ ω(z) implies that Φ(S(t)v) = l =
Φ(v), for any t ≥ 0 and v ∈ ω(z). Thus, v ∈ E.
Since αuz (z) is a nonincreasing intersection of nonempty compact sets, it is a nonempty
compact set. The attractivity property is proved at once by a contradiction argument
using the compactness of the closure of uz (G− ). The invariance property follows from
Remarks 2.10 (ii). The fact that αuz (z) ⊂ E is proved like above. Finally, by Remark 2.16 (ii), αuz (z) is invariantly connected. Since αuz (z) ∈ E, αuz (z) is connected.
Remarks 4.3.
1) Assume that the hypotheses of Proposition 4.2 hold. The above assertions i) and ii)
simply mean that ω(z) ⊂ El , where El = {v ∈ E | Φ(v) = l}. If El is discrete, it follows
from Lemma 2.9 that there exists only one element v0 ∈ El such that ω(z) = v0 , that
is, that S(t)z → v0 as t → +∞; and the orbit through z is said to be convergent. In
general, if the sets El are not discrete, the positive orbits are not convergent (for examples, see [PaMe] in the finite-dimensional case or [PoRy] in the infinite-dimensional
case). However, in the frame of Example 2.2, the positive orbit through z is shown to
be convergent if there is v ∈ ω(z) such that the spectrum of the linear map Σv (1) intersects the unit circle in C at most at the point 1 and 1 is a simple eigenvalue ([HR92b],
[BrP97b]), where Σv (t) is the C 0 -semigroup generated by the linear operator A +Df (v).
Below, we shall give some examples of reaction-diffusion equations and damped wave
- 65 equations, for which the convergence property holds even if the sets El are not discrete
(see [Po00] for a review of convergence results).
2) Assume that S(t) is an asymptotically smooth gradient system, which has the property that, for any bounded set B ⊂ X, there exists τ ≥ 0 such that γτ+ (B) is bounded.
Then, S(t) is point dissipative if and only if E is bounded.
3) More generally, let S(t) be a continuous semigroup and Φ be a Lyapunov functional
associated with the semigroup S(t). Let H = {x ∈ X | Φ(S(t)x) = Φ(x) , ∀t ≥ 0} and
M be the maximal invariant subset of H. One shows like in the proof of Proposition 4.2
that, if z ∈ X is such that γ + (z) is relatively compact in X, then the ω- limit set ω(z)
is contained in M .
4) The simplest example of a gradient system, from which the term “gradient” actually
comes, is the O.D.E. ẏ = ∇F (y), where F ∈ C 1,1 (Rn , R). The associated strict Lyapunov functional is Φ(y) = −F (y). If F is an analytic function, the bounded orbits are
convergent ([Loj]).
For a continuous semigroup, the definitions of stable and unstable sets are analogous
to those given for a discrete system in (3.13). We recall that
W u (E) ≡ W u (E, S(t)) = {v ∈ X | there exists a negative orbit uv of S(t)
such that uv (0) = v and δX (uv (−t), E) →t→+∞ 0} ,
W u (e) ≡ W u (e, S(t)) = {w ∈ X | there exists a negative orbit uw of S(t)
such that uw (0) = w and uw (−t) →t→+∞ e} ,
W s (e) ≡ W s (e, S(t)) = {v ∈ X | S(t)v →t→+∞ e} ,
where e is any element of E.
Two important remarks should be made about the sets W u (e, S(t)) and W s (e, S(t)).
Remark 4.4. If S(t) is a continuous semigroup, then, for any t0 > 0, e is a fixed point
of S(t0 ) and we have the equalities
W u (e, S(t)) = W u (e, S(t0 )) ,
W s (e, S(t)) = W s (e, S(t0 )) .
Remark 4.5. Assume that X is a Banach space. We recall that a continuous semigroup S(t) is said to be of class C r , r ≥ 1, if, for any t ∈ G+ , S(t) ∈ C r (X, X). An
- 66 equilibrium point e of a continuous semigroup S(t), t ≥ 0, is hyperbolic if the linear
map De S(t)t=1 = DS1 (e), where S1 = S(1) satisfies:
σ(DS1 (e)) ∩ {z ∈ C | |z| = 1} = ∅ .
(4.3)
The condition (4.3) means that e is a hyperbolic fixed point of S1 . We set L = DS1 (e).
Let σ+ = {z ∈ σ(L) | |z| > 1} and σ− = {z ∈ σ(L) | |z| < 1}. If σ+ is a finite set of
ne elements, then ne is called the index ind(e) of e. Let P+ and P− be the spectral
projections corresponding to the sets σ+ and σ− , let X± = P± X and L± = LP± .
There exist two small positive numbers δ± such that sup{|z| | z ∈ σ− } < 1 − 2δ− and
sup{|z| | z ∈ σ(L−1
+ )} < 1 − 2δ+ . For any u+ ∈ X+ , u− ∈ X− , we set
kL−n
+ u+ k X
,
n
n≥0 (1 − δ+ )
ku+ k1 = sup
kLn− u− kX
,
n
n≥0 (1 − δ− )
ku− k1 = sup
and, for u ∈ X, we introduce the norm k · k1 on X, defined by
kuk1 = sup(kP+ uk1 , kP− uk1 ) ,
(4.4)
which is equivalent to the original norm k · kX .
±
= OR (0) ∩ X± . One shows that
For any R > 0, let OR (e) = {y ∈ X | ky − ek1 ≤ R}, OR
+
−
−
+
there exist a positive number R, two functions g+ ∈ C 1 (OR
, OR
), g− ∈ C 1 (OR
, OR
),
such that the local stable and unstable manifolds of S1 at e are given by
−
}
W s (e, S1 , OR (e)) = {u ∈ X | u = e + P− (u − e) + g− (P− (u − e)) , P− (u − e) ∈ OR
+
}.
W u (e, S1 , OR (e)) = {u ∈ X | u = e + P+ (u − e) + g+ (P+ (u − e)) , P+ (u − e) ∈ OR
(4.5)
The functions g± satisfy g± (0) = 0 and Dg± (0) = 0. If the index ind(e) is finite, then
W u (e, S1 , OR (e)) is a manifold of dimension ind(e).
Furthermore, there exist positive constants Re , Ke and βe such that, if S1n y ∈ ORe (e),
for n = 1, . . . , m, then,
δX (S1n y, W u (e, S1 , ORe (e))) ≤ Ke exp(−βe n)δX (y, W u (e, S1 , ORe (e))) ,
1≤n≤m,
(4.6)
where βe depends on δ± (see [Wel], [BV83], [ChHaTa], for example).
Assume now that S(t) is a semigroup of class C 1 on the Banach space X and a gradient
system with a Lyapunov functional Φ satisfying, for any t > 0,
Φ(S(t)x) < Φ(x) ,
∀x ∈ X ,
x∈
/E .
(4.7)
- 67 Suppose also that e is a hyperbolic equilibrium point of S(t) with finite index ind(e).
Then one can show that, for any ρ > 0, there exists a positive number r such that
W u (e, S(t)) ∩ BX (e, r) ⊂ W u (e, S1 , Oρ (e)) ,
(4.8)
where BX (e, r) = {x ∈ X | kx − ekX < r}.
Often, it is easier to construct and study invariant manifolds of time-one maps
rather than those of the flow S(t). Remark 4.4 and Remark 4.5 show that, in the case
of continuous gradient systems with a Lyapunov functional satisfying the condition
(4.7), the local and global unstable manifolds of S(t) at e can indeed be replaced by
those of the map S1 .
From Theorem 2.26 and Proposition 4.2, one easily deduces the following result:
Theorem 4.6. Let S(t), t ∈ G+ , be an asymptotically smooth gradient system, which
has the property that, for any bounded set B ⊂ X, there exists τ ≥ 0 such that γτ+ (B)
is bounded. If the set of equilibrium points E is bounded, then S(t) has a compact
global attractor A and A = W u (E).
S
Furthermore, if E is a discrete set, E is a finite set {e1 , e2 , . . . en0 } and A = ej ∈E W u (ej ).
If E is a discrete set and uz ∈ C 0 (G+ , A) is a complete orbit in A through z, there
exist equilibrium points ej and ek such that αuz (z) = ej and ω(z) = ek . If z is not an
equilibrium point, Φ(ek ) < Φ(a) ≤ Φ(ej ). The orbit which joins the points ej and ek is
called a heteroclinic orbit.
Under the hypotheses of Theorem 4.6, we introduce the m0 distinct values v1 >
v2 > . . . > vm0 of the set {Φ(e1 ), Φ(e2 ), . . . , Φ(en0 )} and let
E j = {eji ∈ E | Φ(eji ) = vj } , j = 1, . . . , m0 .
The sets E 1 , E 2 , ....., E m0 define a Morse decomposition of the attractor A, i.e.,
(i) the subsets E j are compact, invariant and disjoint;
S
(ii) for any a ∈ A \ j E j and every complete orbit ua through a, there exist k and l,
depending on ua , so that k < l, αua (a) ∈ E k and ω(a) ∈ E l .
For 1 ≤ k ≤ m0 , one defines
k
A =
[
u
{W (e) | e ∈
m0
[
j=k
Ej} ,
- 68 and, for 0 < d < d0 ≡ inf 2≤k≤m0 (vk−1 − vk ),
Fdk = {u ∈ X | Φ(u) ≤ vk−1 − d} ,
where v0 is choosen so that v0 > v1 +d0 . Assume now that the hypotheses of Theorem 4.6
hold and that E is a discrete set. Then, the same arguments as those used to prove
Theorem 4.6, show that, for any 0 < d < d0 and any k, 1 ≤ k ≤ m0 , Ak is the (compact)
global attractor of S(t)/F k .
d
If X is a Banach space and all the equilibrium points of S(t) are hyperbolic, then,
using the above Morse decomposition and Remark 4.5, one shows that the global attractor exponentially attracts a neighbourhood of it. This property plays an important
role in the lower semicontinuity of global attractors. Its proof is implicitely contained
in [HR89] and can be found in [BV89a] (See also [Hal88], [BV89b], [Ko90] and [GR00]).
Theorem 4.7. Let X be a Banach space. Assume that the hypotheses of Theorem 4.6
hold and that the Lyapunov functional satisfies (4.7), for any t ∈ G+ , t 6= 0. Suppose
moreover that, either S(t) is a continuous semigroup of class C 1 or S is a C 1 -mapping
from X to X and that, in both cases, all the equilibrium points of S(·) are hyperbolic.
Then, there exist a bounded neighbourhood B1 of the global attractor A and positive
constants C1 , d1 < d0 , γ, such that,
δX (S(t)(F̃d11 ), A) ≤ C1 exp(−γt) ,
∀t ∈ G+ ,
(4.9)
where F̃d11 = Fd11 ∩ B1 .
The number γ > 0 in (4.9) depends on the minimum, over e ∈ E, of the distance of
the spectrum of DS1 (e) to the unit circle in C. The C 1 -regularity hypotheses in Theorem 4.7 and in Remark 4.5 can be weakened and replaced by the following assumption
S(1)(y + e) = e + Ly + Q(e, y), where L ∈ L(X, X) satisfies the spectral hypothesis
(4.3), Q(e, 0) = 0 and Q(e, ·) : X → X is Lipschitz-continuous on the bounded sets
of X. We also assume that the Lipschitz constant of Q(e, ·) on the balls BX (0, r) is a
continuous, nondecreasing function of r, vanishing at r = 0. In this case, the mappings
g ± of Remark 4.5 are only Lipschitzian mappings.
The proof of Theorem 4.7 actually shows that, for any u0 ∈ F̃d11 , there exists a finite
number of trajectories S(t)u0j , u0j ∈ A, t ∈ [tj , tj+1 ), j = 0, .., k(u0 ), with t0 = 0
and tk(u0 ) = +∞ such that kS(t)u0 − S(t)u0j kX ≤ C1 exp(−γt), for any t ∈ [tj , tj+1 ).
- 69 k(u )−1
t
The “trajectory” ũ(t) = ∪j=00 (∪tj+1
S(t)u0j ) is called a finite-dimensional combined
j
trajectory by Babin and Vishik in [BV89a]. Actually, under additional conditions, it
is shown there, that, for any η > 0, one can construct a finite-dimensional combined
trajectory ũη (t) ∈ A such that kS(t)u0 − ũη (t)kX ≤ C(η) exp(−ηt), where C(η) > 0
depends on η. This trajectory ũη (t) belongs piecewise to invariant manifolds, whose
dimension increases when η decays to zero.
The assumption (4.7) implies that, for any hyperbolic equilibrium point e ∈ E,
there is a neighbourhood Ue of e such that, if x0 ∈ Ue \W s (e, S(t)), then there exists
t0 = t0 (x0 ) > 0 so that S(t)x0 ∈
/ Ue , for t ≥ t0 , that is, S(t)x0 eventually leaves Ue
never to return. This property plays an important role in the proof of (4.9) and also in
the proof of the following theorem, stating that the unstable and stable manifolds of a
hyperbolic equilibrium point are embedded submanifolds of X.
Theorem 4.8. Assume that the hypotheses of Theorem 4.7 hold and that S1 = S(1)
as well as the linear map DS1 (y) are injective at each point y of the global attractor
A, then, for each e ∈ E, the unstable set W u (e, S(t)) is an embedded C 1 -submanifold
of X of finite dimension equal to ind(e), which implies that the Hausdorff dimension
dimH (A) is finite and equal to maxe∈E ind(e). If furthermore, for each e ∈ E, S1 is
injective and DS1 (y) has dense range at each point y of W s (e, S(t)), then the stable set
is an embedded C 1 -submanifold of X of codimension ind(e).
The above theorem is a consequence of Remark 3.7, Remark 4.4 and [He81, Theorem 6.1.9, Theorem 6.1.10] (see also [BV83]). Under the hypotheses of Theorem 4.8,
one shows that, for any hyperbolic equilibrium point e, there is a neighbourhood Qe of
W s (e, S) such that, if x0 ∈ Qe \W s (e, S(t)), then there exists t0 = t0 (x0 ) > 0 so that
S(t)x0 ∈
/ Qe , for t ≥ t0 .
Remarks.
1. Assume that the gradient system S(t) is generated by the evolutionary equation (2.1).
Under additional hypotheses, Theorem 4.8 implies that the Hausdorff dimension of the
global attractor of (2.1) is estimated by the maximum of the number of eigenvalues with
positive real part of A + Df (e), e ∈ E. If in (2.1), f is replaced by λf , one thus obtains
asymptotic estimates of dimH (A), when λ → +∞, by using asymptotic estimates of the
number of positive eigenvalues of the operators A + λDf (e) (see [BV89b, Chapter 10,
Section 4]).
- 70 2. If S(t) is generated by an evolutionary equation, the injectivity of S1 (resp. DS1 )
comes usually from a backward uniqueness result of the solutions of the evolutionary
equation (resp. the corresponding linearized equation). And one shows that the range
of DS1 is dense by proving that the adjoint map is injective. Backward uniqueness
results are known to hold for a large class of parabolic equations (see [BT], [He81,
Chapter 7, Section 6], [Gh86]). The hyperbolicity of the equilibrium points is usually
a generic property with respect to the various parameters involved in the definition of
the semigroup S(t).
Gradient systems depending on parameters.
As in Section 3.1, we consider a family of semigroups Sλ (t) : X → X, t ∈ G+ ,
depending on a parameter λ ∈ Λ, where Λ = (Λ, dλ ) is a metric space.
Let λ0 be a nonisolated point of Λ. We assume that Sλ0 (t) is a gradient system satisfying
the hypotheses of Theorem 4.7, the conditions (H.1b) and (3.8) of Section 3.1, then
Proposition 3.1, Proposition 3.3 and Theorem 4.7 imply that, for λ close enough to λ0 ,
we have,
γγ0
(4.10)
δX (Aλ , Aλ0 ) ≤ cδΛ (λ, λ0 ) γ+β0 ,
for some positive constant c.
We now turn to lower semicontinuity results and estimates of the semi-distance
δX (Aλ0 , Aλ ). Lower semicontinuity properties have been first proved in a very general
setting in [BV86] and [HR89] (see also [Hal88, Chapter 4, Section 10]). To show some
of the ingredients, which are necessary, we begin with a very simple result. Hereafter,
we denote by Eλ the set of equilibria of Sλ . Besides the condition (H.1b) of Section 3.1,
we introduce the following hypotheses:
(H.2) The set Eλ0 is a finite set, say Eλ0 = {eλ1 0 , . . . , eλn0λ };
0
(H.3) the global attractor Aλ0 is compact and,
nλ0
Aλ0 =
[
j=1
nλ0
W
u
(eλj 0 , Sλ0 (t))
=
[
W u (eλj 0 , Sλ0 (1)) ;
j=1
and
(H.4) for j = 1, . . . , nλ0 , there exists a neighbourhood Oj of eλj 0 in X such that
lim δX (W u (eλj 0 , Sλ0 (1), Oj ), Aλ ) = 0 .
λ→λ0
- 71 Proposition 4.9. If the hypotheses (H.1b), (H.2), (H.3) and (H.4) hold, the global
attractors Aλ are lower semicontinuous at λ = λ0 , i.e., δX (Aλ0 , Aλ ) → 0 as λ → λ0 ,
which, together with Proposition 3.1, implies that the attractors Aλ are continuous at
λ = λ0 .
Proof. We give the proof only when G+ = [0, +∞). Without loss of generality, we
suppose that t0 = 1 in the hypothesis (H.1b).
Assume that the sets Aλ are not lower semicontinuous at λ = λ0 . Then, there exist
ε > 0 and, for any m ∈ N, an element λm ∈ NΛ (λ0 , 1/m) and an element ϕm ∈ Aλ0
such that δX (ϕm , Aλm ) > ε. Since Aλ0 is compact, the sequence ϕm converges to an
element ϕ0 ∈ Aλ0 and
δX (ϕ0 , Aλm ) > ε/2 ,
∀m ∈ N ,
m ≥ m0 ,
(4.11)
where m0 ∈ N, m0 > 0.
If ϕ0 ∈ W u (eλj 0 , Sλ0 (1), Oj ), for some j = 1, . . . , nλ0 , (4.11) contradicts the hypothesis
(H.4). Thus, assume that ϕ0 ∈
/ W u (eλj 0 , Sλ0 (1), Oj ), for any j = 1, . . . , nλ0 , then there
are n0 and ψ0 ∈ W u (eλi 0 , Sλ0 (1), Oi), for some i, so that ϕ = Sλ0 (n0 )ψ0 . Since Aλ0
is compact, for any ε > 0, there exists δ > 0, depending only on Aλ0 , such that, if
δX (v, ψ0 ) ≤ δ, then,
δX (Sλ0 (nϕ )v, Sλ0 (nϕ )ψ0 ) ≤ ε/4 .
(4.12)
The hypotheses (H.4) and (H.1b) imply that we can choose θ = θ(ε, nϕ ) > 0, such that,
for λ ∈ NΛ (λ0 , θ), there exists vλ ∈ Aλ so that
δX (ψ0 , vλ ) ≤ δ ,
δX (Sλ (nϕ )vλ , Sλ0 (nϕ )vλ ) ≤ ε/4 .
(4.13)
We deduce from (4.12) and (4.13) that, for λ ∈ NΛ (λ0 , θ),
δX (Sλ (nϕ )vλ , ϕ) ≤ ε/2 ,
which contradicts the property (4.11).
Remark 4.10. Usually, when X is a Banach space, the hypothesis (H.4) is shown by
proving that, for λ close enough to λ0 , Sλ (t) has, at least, nλ0 equilibrium points eλj ,
j = 1, . . . , nλ0 and that, for each j = 1, . . . , nλ0 , there exists a neighbourhood Ojλ of eλj
such that
(4.14)
lim δX (W u (eλj 0 , Sλ0 (1), Oj ), W u (eλj , Sλ (1), Ojλ )) = 0 .
λ→λ0
- 72 If, for instance, the equilibrium points eλj 0 are all hyperbolic and that the mapping Sλ (1)
converges to Sλ0 (1) in C 1 (B, X), as λ → λ0 , where B is a bounded neighbourhood of
Aλ0 , then the property (4.14) holds (see [Wel] and [BV86], for example). In the general
case, if eλj 0 is not hyperbolic, the hypothesis (H.4) is much more difficult to prove.
However, in a particular regular perturbation of the Chafee-Infante equation, Kostin
[Ko95] proved this condition.
If we want to estimate the semi-distance δX (Aλ0 , Aλ ), we need stronger hypotheses.
Actually, general results are only known in the case where Sλ0 is a gradient system. The
next theorem summarizes the above discussion and gives an estimate of the distance
HdistX (Aλ , Aλ0 ). To simplify, we do not state the optimal hypotheses (for a more
general statement and various applications to perturbed problems, see [HR89], [Hal88],
[Ko90], [Ra95]).
Theorem 4.11. Let be given a family of semigroups Sλ , λ ∈ Λ, on a Banach space X,
whith compact global attractors Aλ . Assume that the hypothesis (H.1b) holds, that
Sλ0 is a gradient system with a Lyapunov functional satisfying (4.7). Suppose moreover
that, either Sλ0 (t) is a continuous semigroup of class C 1 or Sλ0 is a C 1 -mapping from X
to X, and that all the equilibrium points of Sλ0 (·) are hyperbolic.
1. If Sλ (1) converges to Sλ0 (1) in C 1 (B1 , X), where B1 is a bounded neighbourhood of
Aλ0 , then the global attractors Aλ are continuous at λ = λ0 .
2. If, moreover, there are positive constants C and p such that
kSλ0 (1) − Sλ (1)kC1 (B1 ,X) ≤ CδX (λ, λ0 )p ,
(4.15)
for any λ in some neighbourhood of λ0 , then there exist a neighbourhood NΛ (λ0 , ε),
two positive constants Ĉ and p̂, 0 < p̂ ≤ p, such that, for λ ∈ NΛ (λ0 , ε),
HdistX (Aλ , Aλ0 ) ≤ ĈδX (λ, λ0 )p̂ .
(4.16)
Theorem 4.11 gives good information about the size of the compact global attractor
Aλ for λ near λ0 . Under the condition of hyperbolicity of the equilibria, the size of the
global attractor does not change. However, the flows may not stay the same for each
λ. As in Section 3.1, one gets more precise information, when Sλ0 (t) is a Morse-Smale
semigroup. In the context of gradient systems, Theorem 3.10 implies the following
statement, which is very useful in the applications.
- 73 Theorem 4.12. Let be given a family of semigroups Sλ , λ ∈ Λ on a Banch space X,
whith compact global attractors Aλ . Assume that the hypothesis (H.1b) holds, that,
for any λ, Sλ is a gradient system with Lyapunov functional satisfying (4.7). Suppose
moreover that, either, for any λ, Sλ (t) is a continuous semigroup of class C 1 or Sλ0 is a
C 1 -mapping from X to X and that Sλ (1) converges to Sλ0 (1) in C 1 (B1 , X), where B1
is a bounded neighbourhood of Aλ0 . In addition, suppose that:
1) Sλ (1) and DSλ (1)(y) are injective at each point y of Aλ , λ ∈ Λ,
2) every equilibrium point eλ0 ∈ Eλ0 is hyperbolic and W u (eλ0 ,1 , Sλ0 (1)) is transversal
s
to Wloc
(eλ0 ,2 , Sλ0 (1)), for any equilibria eλ0 ,1 , eλ0 ,2 ∈ Eλ0 ,
then, there exist n0 ∈ N and a neighbourhood NΛ (λ0 , ε) of λ0 , such that, for any
λ ∈ NΛ (λ0 , ε), Sλ (t) is a Morse-Smale system (i.e. Sλ (n0 ) is a Morse-Smale map) and
Sλ (n0 ) is conjugate to Sλ0 (n0 ).
In the remaining parts of this section, we are going to describe classical examples
of gradient systems generated by evolution equations.
- 74 4.2. Retarded functional differential equations
In many problems in physics and biology, the future state of a system depends not only
on the present state, but also on past states of the system. The theory of functional
differential equations probably started with the work of Volterra [Vo28] [Vo31], who, in
his study of models in viscoelasticity and population dynamics, introduced some rather
general differential equations incorporating the past states of the system. Since then,
retarded functional differential equations (RFDE’s) play an important role in biology
(predator-prey models, spread of infections, circummutation of plants, etc . . .) and in
mechanics. Here, we mainly describe a model RFDE arising in viscoelasticity. For
further studies and generalizations to neutral functional differential equations, we refer
the reader to the book of Hale and Lunel [HVL] as well as to [Hal88] and [Nu00].
For a given δ > 0 and n ∈ N\{0}, let C = C 0 ([−δ, 0); Rn ) be the space of continuous
functions from [−δ, 0] into Rn equipped with the norm k · k = k · kC . For any α ≥ 0,
for any function x : [−δ, α) → Rn and any t ∈ [0, α), we let xt denote the function from
[−δ, 0] to Rn defined by xt (θ) = x(t + θ), θ ∈ [−δ, 0].
Suppose that f ∈ C k (C, Rn ), k ≥ 1, and that f is a bounded map in the sense that f
takes bounded sets into bounded sets. An autonomous retarded functional differential
equation with finite delay is a relation
ẋ(t) = f (xt ) ,
(4.17)
where ẋ(t) is the right hand derivative of x(t) at t.
For a given ϕ ∈ C, one says that x(t, ϕ) is a solution of (4.17) on the interval [0, αϕ),
αϕ > 0, with initial value ϕ at t = 0, if x(t, ϕ) is defined on [−δ, αϕ ), satisfies (4.17) on
[0, αϕ ), xt (·, ϕ) ∈ C for t ∈ [0, αϕ ) and x0 (·, ϕ) = ϕ. Using the contraction fixed point
theorem, one shows that, for any ϕ ∈ C, there exists a unique mild solution x(t, ϕ)
defined on a maximal interval [−δ, αϕ ). Moreover, x(t, ϕ) is continuous in (t, ϕ), of
class C k in ϕ and, for t ∈ (kδ, αϕ ), of class C k in t. If αϕ < +∞, then kxt (·, ϕ)k → +∞
as t → α−
ϕ.
We assume now that all the solutions of (4.17) are defined for t ∈ [0, +∞). Then,
the one-parameter family of maps S(t), t ≥ 0 on C defined by S(t)ϕ = xt (·, ϕ) is a
continuous semigroup on C. We also introduce the linear semigroup V (t) : C → C,
t ≥ 0, given by
(V (t)ϕ)(θ) = ϕ(t + θ) − ϕ(0) , t + θ < 0 ,
(4.18)
= 0 , t+θ ≥0 .
- 75 The following theorem states the basic qualitative properties of the semigroup S(t) and
can be found in [HVL], for example.
Theorem 4.13. If the positive orbits of bounded sets are bounded, then S(t) is a
compact map for t ≥ δ. Moreover, for t ≥ 0, one has
S(t)ϕ = U (t)ϕ + V (t)ϕ ,
(4.19)
where U (t) is a compact map from X into X for t ≥ 0 and V (t) has been defined
in (4.18). Furthermore, for any β > 0, there is an equivalent norm | · | on C so that
|S(t)ϕ| ≤ exp(−βt)|ϕ|, t ≥ 0, and S(t) is an α-contraction in this norm for t ≥ 0.
The fact that S(t) can be written as (4.19) had been remarked by Hale and Lopes
[HaLo]. The next result is mainly a consequence of Theorem 2.26, Theorem 2.38 and
Theorem 4.13. The analyticity property in the third statement is due to [Nu73].
Theorem 4.14. If the positive orbits of bounded sets are bounded and if S(t) is point
dissipative, then
(i) S(t) has a connected compact global attractor A ⊂ C;
(ii) there is at least an equilibrium point (a constant solution) of (4.17);
(iii) if f ∈ C k (C, Rn ), k ≥ 0, (resp. analytic), then any element u of the attractor A is
a C k+1 function (resp. analytic);
(iv) if f is analytic, then S(t) is one-to-one on A.
We now present an example of a gradient system generated by a RFDE, which
arises in viscoelasticity [LN]. We now let δ = 1 and suppose that b is a function in
C 2 ([−1, 0], R), such that b(−1) = 0, b(s) > 0, b′ (s) ≥ 0, b′′ (s) ≥ 0 and that
b′′ (θ0 ) > 0 for some θ0 ∈ (−1, 0) .
(4.20)
Let g ∈ C 1 (R, R) be such that
G(x) ≡
Z
x
0
g(s) ds → +∞ as |x| → +∞ .
(4.21)
We consider the equation
ẋ(t) = −
Z
0
−1
b(θ)g(x(t + θ)) dθ .
(4.22)
- 76 R0
Equation (4.22) is a special case of (4.17) with f (ϕ) = − −1 b(θ)g(ϕ(θ)) dθ. Let S(t) be
the local semigroup on C defined by S(t)ϕ = xt (·, ϕ), where x(t, ϕ) is the local solution
of (4.22) through ϕ at t = 0. To show that the solutions of (4.22) exist globally, one
introduces the functional on C
1
Φ(ϕ) = G(ϕ(0)) +
2
We set Lθ (ψ) =
R0
θ
Z
0
−1
′
b (θ)
0
Z
g(ϕ(s)) ds
θ
2
dθ .
g(ψ(s)) ds. A short computation shows that, for t ≥ 0,
1
d
(Φ(S(t)ϕ)) = − b′ (−1)[L−1 (S(t)ϕ)]2 +
dt
2
Z
0
−1
b′′ (θ)[Lθ (S(t)ϕ)]2 dθ .
(4.23)
The hypotheses on b and g imply that Φ(ϕ) → +∞ as kϕk → +∞ and, due to (4.23),
that Φ(S(t)ϕ) ≤ Φ(ϕ), for t ≥ 0. Therefore, the solutions of (4.22) exist globally and
the orbits of bounded sets are bounded. The next theorem summarizes the properties
of the semigroup S(t) ([Hal88], [HRy], [Hal00]).
Theorem 4.15. Assume that the conditions (4.20) and (4.21) hold. Then, the ω-limit
set of any orbit is a single equilibrium point.
If moreover, the set E of the zeros of g is bounded, the semigroup S(t) generated by
(4.22) is a continuous gradient system and admits a compact connected global attractor
Abg in C. If, in addition, each element of E is hyperbolic, then dim W u (x0 ) = 1, for any
S
x0 ∈ E and Abg = x0 ∈E W u (x0 ).
Sketch of the proof. We first observe that any solution x(t) of (4.22) satisfies
′
ẍ(t) + b(0)(g(x(t)) = b (−1)L−1 (xt ) +
Z
0
b′′ (θ)Lθ (xt ) dθ .
(4.24)
−1
Suppose that Φ(S(t)ϕ) = Φ(ϕ), for t ≥ 0, then (4.23) and (4.24) imply that the solution
x(t) through ϕ satisfies
ẍ(t) + b(0)(g(x(t)) = 0 ,
together with Ls (xt ) = 0 for s in some interval I0 containing θ0 . It follows that ẋ is a
constant. The boundedness of x(t) implies that x(t) is a constant and thus ϕ is a zero
of g. Hence Φ is a strict Lyapunov functional. The existence of the compact global
attractor Abg is a direct consequence of Theorem 4.6 and Theorem 4.14.
- 77 R0
If g(c) = 0, the linear variational equation about c is ẏ(t) = − −1 b(s)g ′ (c)y(t + s) ds
and the eigenvalues λ of the linear variational equation are given by
λ = −
Z
0
b(s)g ′ (c) exp(λs) ds .
−1
It is possible to show that the equilibrium point c is hyperbolic if and only if g ′ (c) 6= 0.
If g ′ (c) = 0, λ = 0 is a simple eigenvalue. Property 1) of Remarks 4.3 then implies that
the ω-limit set of any positive orbit is a singleton. If g ′ (c) > 0, then c is stable. Finally,
one easily shows that, if g ′ (c) < 0, then c is unstable with dim W u (c) = 1.
Suppose now that b is a fixed function and consider the global attractor Abg as a
function of the parameter g. Semicontinuity and continuity results of Abg with respect
to g are proved in [HR89] and are actually an application of Proposition 4.9 and Theorem 4.11. We have seen that, for each zero c with g ′ (c) < 0, c is an unstable equilibrium
point with dim W u (c) = 1; this means that there are two distinct complete orbits ϕe (t)
and ψe (t) which approach e as t → −∞. Since the ω-limit set of ϕe (t) (resp. ψe (t)) is
a single equilibrium point eϕ (resp. eψ ), the next problem is to determine if eϕ (resp.
eψ ) is smaller or larger than e. If E = {e1 , e2 , e3 } with e1 < e2 < e3 , the flow on Abg
preserves the natural order since Abg is connected. The case when E = {e1 , e2 , e3 , e4 , e5 }
with e1 < e2 < e3 < e4 < e5 , has been studied by Hale and Rybakowski [HRy]. To
state their result, it is convenient to use the notation j[k, l] to mean that the unstable
point ej is connected to ek and el by a trajectory. If g has five simple zeros, then e2 , e4
are unstable, while e1 , e3 , e5 are stable.
Theorem 4.16. Let b be fixed. One can realize each of the following flows on Abg by
an appropriate choice of g with five simple zeros:
(i) 2[1, 3], 4[3, 5];
(ii) 2[1, 4], 4[3, 5];
(iii) 2[1, 5], 4[3, 5];
(iv) 2[1, 3], 4[2, 5];
(v) 2[1, 3], 4[1, 5];
- 78 4.3. Scalar parabolic equations
The simplest and most studied gradient partial differential equation is the semilinear heat or reaction-diffusion equation, which models several physical phenomena like
heat conduction, population dynamics, etc...The heat equation belongs to the class of
parabolic equations, where smoothing effects take place in finite positive time. Here,
we study this equation under very simplified hypotheses on the nonlinearity.
Let Ω be a bounded domain in Rn , with Lipschitzian boundary. We consider the
following heat equation
∂u
(x, t) = ∆u(x, t) + f (u(x, t)) + g(x) ,
∂t
u(x, t) = 0 , x ∈ ∂Ω , t > 0 ,
u(x, 0) = u0 (x) ,
x∈Ω,
t>0,
(4.25)
x∈Ω,
where g is in L2 (Ω) and f : R → R is a locally Lipschitz continuous function. We
introduce the operator A = −∆D with domain D(A) = {v ∈ H01 (Ω) | − ∆v ∈ L2 (Ω)}
and set V = H01 (Ω) ≡ D(A)1/2 .
In the case n ≥ 2, we assume that the locally Lipschitz continuous function f also
satisfies the following growth condition:
(A.1) there exist positive constants C0 and α, with (n − 2)α ≤ 2 such that
|f (y1 ) − f (y2 )| ≤ C0 (1 + |y1 |α + |y2 |α )|y1 − y2 | ,
∀y1 , y2 ∈ R .
(4.26)
The restriction (n − 2)α ≤ 2 has been made only for sake of simplicity. Most of the
results of Section 4.3 also hold if (n − 2)α ≤ 4 (see Remark 4.18 1. below). The
hypothesis (A.1) together with the Sobolev embeddings properties, allow to define the
mapping u ∈ V 7→ f (u) ∈ L2 (Ω), by (f (u))(x) = f (u(x)), for almost every x ∈ Ω.
This mapping is Lipschitzian on the bounded sets of V . With the above definitions of
A and V , we can rewrite (4.25) as an abstract evolutionary equation in V :
du
(t) = −Au(t) + f (u(t)) + g ,
dt
t>0,
u(0) = u0 ,
(4.27)
Since A is a sectorial operator and f : V → L2 (Ω) is Lipschitzian on the bounded sets of
V , for any r ≥ 0, there exists T ≡ T (r) > 0 such that, for any u0 ∈ V , with ku0 kV ≤ r,
the equation (4.27) has a unique classical solution u ∈ C 0 ([0, T ], V ) ∩ C 1 ((0, T ], L2(Ω)) ∩
- 79 C 0 ((0, T ], D(A)).
Later, when we need more regularity on f , we suppose in addition that
(A.2) f ∈ Cu1 (R, R), f ′ is locally Hölder continuous and, if n ≥ 2, there exist
nonnegative constants C1 , α1 , β1 , such that
|f ′ (y1 ) − f ′ (y2 )| ≤ C1 (1 + |y1 |β1 + |y2 |β1 )|y1 − y2 |α1 ,
∀y1 , y2 ∈ R ,
(4.28)
where α1 > 0, (α1 + β1 )(n − 2) ≤ 2 if n ≥ 2. In this case, f is a C 1,α1 -mapping from V
into L2 (Ω).
Remark.
For sake of simplicity, we have provided the above heat equation with homogeneous
Dirichlet boundary conditions. All the assertions of this subsection remain true if we
replace them in (4.25) by homogeneous Neumann conditions, in which case V = H 1 (Ω).
Even, much more general boundary conditions may be chosen. Furthermore, we can
Pi,j=n ∂
∂
(aij (x) ∂x
)+
replace the Laplacian operator by any second order operator i,j=1 ∂x
i
j
a0 (x), where aij , a0 are smooth enough functions of x and the matrix [aij (x)]i,j is symmetric, positive definite, for any x ∈ Ω.
To obtain global existence of the solutions of (4.27), we need to impose, for example,
a dissipation condition. Here we assume that
(A.3) there exist constants C2 ≥ 0 et µ ∈ R such that
yf (y) ≤ C2 + µy 2 ,
1
F (y) ≤ C2 + µy 2 , ∀y ∈ R ,
2
(4.29)
with
µ < λ1 ,
(4.30)
where λ1 is the first eigenvalue of the operator A and where F is the primitive F (y) =
Ry
f (s) ds of f . Global existence of solutions of (4.25) already holds under the hypoth0
esis (4.29). Condition (4.30) will ensure that all the solutions are uniformly bounded.
Indeed, introducing the functional Φ0 ∈ C 0 (V, R) given by
Φ0 (u) =
Z Ω
1
|∇u(x)|2 − F (u(x)) − g(x)u(x)
2
dx ,
(4.31)
- 80 one shows that, if u(t) is a classical solution of (4.27) for t ≤ T , then Φ0 (u(t)) ∈
C 0 ([0, T ]) ∩ C 1 ((0, T ]) and
d
Φ0 (u(t)) = −kut (t)k2L2 ,
dt
∀t ∈ (0, T ] ,
(4.32)
which implies that Φ0 is a strict Lyapunov function of (4.27). Moreover, using the
assumptions (A.1), (A.3) and the property (4.32), we obtain, for any 0 < ε < λ1 − µ,
µ+ε
1
1
(1 −
)k∇u(t)k2L2 − C|Ω|− kgk2L2 ≤ Φ0 (u(t)) ≤ Φ0 (u(0))
2
λ1
2ε
1
≤ C ∗ (1 + kgkL2 + ku(0)k1+α
H 1 )ku(0)kH ,
(4.33)
where C, C ∗ are positive constants. This implies that all the solutions of (4.27) are
global and the orbits of bounded sets are bounded. We notice that the set EP of the
equilibrium points of (4.27) is given by
EP = {u ∈ V | ∆D u + f (u) + g = 0} .
Due to the dissipative condition (A.3), this set is bounded in V .
If we let S(t)u0 denote the solution u(t) of (4.27), with initial data u0 ∈ V , we have
defined a continuous gradient system S(t) on V . We remark that the map (t, u0 ) 7→
S(t)u0 belongs to C 0 ([0, +∞) × V, V ). Due to a backward uniqueness result of Bardos
and Tartar [BT], the mapping S(t) is injective on V , for any t ≥ 0. Furthermore, since
for any bounded set B ⊂ V , the orbit γ + (B) is bounded in V , one shows, by using the
smoothing properties of S(t), that the orbit γτ+ (B) is bounded in D(A), for any τ > 0
([He81, Theorem 3.5.2]), which implies in particular that S(t) is compact for t > 0.
Applying Proposition 2.5 and Theorem 4.6, one obtains the existence of a compact
global attractor.
Theorem 4.17. Assume that the assumptions (A.1) and (A.3) hold. Then, the semigroup S(t) generated by (4.27) has a compact connected global attractor A = W u (EP )
in V . The semigroup S(t)A is a continuous group of continuous operators. Moreover,
the global attractor A is bounded in D(A) and thus in H 2 (Ω), if the domain Ω is either
convex or of class C 1,1 .
Properties of the compact attractor A
In what follows, we assume that the assumptions (A.1), (A.2) and (A.3) hold.
- 81 First, we recall that A is often bounded in a higher order Sobolev space. If, for instance,
the nonlinearity f belongs to C k (V, L2 (Ω)) (resp. is analytic), then the map (t, u0 ) ∈
[τ, +∞) × V 7→ S(t)u0 ∈ V is of class C k , for τ > 0 (resp. is analytic) (see [He81,
Corollary 3.4.6]). Due to the invariance of A, it follows that S(t)A : t ∈ R 7→ S(t)u0 ∈
V is of class C k (resp. analytic). Arguing by recursion on k, one deduces from the
regularity in time property that, if f ∈ Cbk−j (H j+1 (Ω), H j (Ω)), 1 ≤ j ≤ k, (resp.
C ∞ (H j+1 (Ω), H j (Ω)), for j ≥ 1) and if moreover Ω is of class C k+1,1 (resp. C ∞ ) and
g ∈ H k (Ω) (resp. C ∞ (Ω)), then A is bounded in H k+2 (Ω) (resp. C ∞ (Ω)). If Ω ⊂ Rn ,
n = 1, 2, 3, is either convex or of class C 1,1 , applying for example the results of Section
k
3.3, one shows that, if f ∈ Cbu
(R, R) and f (k) is locally Hölder continuous (respectively
f is analytic), then the map t 7→ S(t)u0 is in C k ((0, +∞), V ) (respectively analytic), for
any u0 ∈ V and thus, that A is bounded in H k+2 (Ω) (resp. C ∞ (Ω)) if moreover Ω is of
class C k+1,1 (resp. C ∞ ).
For regularity in Gevrey spaces, when the Dirichlet boundary conditions are replaced
by periodic ones, we refer to [FT89], [Pr91] and [FeTi98] and the references therein.
The fractal dimension of A is finite. An explicit bound is given in [Te, Chapter
6], for example. By Theorem 3.17, the property of finite number of determining modes
holds.
We remark that e is a hyperbolic equilibrium point of (4.27) if and only if the
spectrum σ(−A + Df (e)) of (−A + Df (e)) does not intersect the imaginary axis in C
or, equivalently here, if 0 is not an eigenvalue of (−A + Df (e)). Moreover, the index
ind(e) is equal to the number of positive eigenvalues lj (e) of −A + Df (e). We recall
that the hyperbolicity of the equilibria of (4.25) is a generic property in g ∈ L2 (Ω)
([BV83], [BV89b, Chapter 6, Theorem 3.4]). Generic hyperbolicity of the equilibrium
points also holds with respect to the domain Ω ([He87]). In the one-dimensional case,
generic hyperbolicity with respect to f has also been proved in [BrCh] and [SW], for
example.
If all of the equilibrium points (e, 0) of (4.27) are hyperbolic, we deduce from TheS
orem 4.6 and Theorem 4.8 that A = ej ∈EP W u (ej ) and that the Hausdorff dimension
dimH (A) is equal to maxe∈EP ind(e). The unstable and stable manifolds W u (e, S(t))
and W s (e, S(t)) are embedded C 1 -submanifolds of V of dimension ind(e) and codimension ind(e), respectively. We shall see later that these stable and unstable manifolds
always intersect transversally if Ω is an interval of R. This property is not known in
higher dimension space. If we allow the function f to depend upon x and assume that
- 82 this function f (x, .) satisfies the assumptions (A.1), (A.2) and (A.3) uniformly in x,
then the resulting semigroup S(t) still admits a compact global attractor. Brunovsky
and Poláčik have proved that the semigroup defined by (4.27) is a Morse-Smale system,
generically in such non linearities f (x, .) (see [BrP97a]). Furthermore, for the unit ball
in R2 , Poláčik has shown that there exists a function f (x, u) for which the tranversality
of the stable and unstable manifolds does not hold (see [Po94]).
In the one-dimensional case, the eigenvalues λi of the operator A fulfill the gap
condition needed in the construction of inertial manifolds. In this case, (4.27) has an
inertial manifold (see [FST]). Mallet Paret and Sell [MPS] have proved that this gap
condition can be replaced by a cone condition, which is less restrictive. As a consequence,
they showed that (4.27) has an inertial manifold of class C 1 , if Ω is either a rectangular
domain (0, 2π/a1 ) × (0, 2π/a2), where a1 , a2 are arbitrary positive numbers or Ω is
the cube (0, 2π)3 and if f : (x, y) ∈ Ω × R 7→ f (x, y) ∈ R is of class C 3 . These
results are valid for the equation (4.25) supplemented with homogeneous Dirichlet or
Neumann boundary conditions or periodic boundary conditions. The existence of an
inertial manifold can also be proved in the case of domains in Rn+1 , which are thin in
n directions ([HR92c], [Ra95]).
If Ω ⊂ R, the positive orbit of every point u0 ∈ V is convergent ([Ze]). If Ω is a
domain in Rn+1 , which is thin in n directions, the positive orbits are still convergent
([HR92b]). In the case n ≥ 1, all the orbits of (4.27) are still convergent if f : R → R
and g are analytic functions ([Si]). For further details on convergence properties, see
[HR92b], [BrP97b] and [Po00].
Remarks 4.18.
1. For sake of simplicity in the exposition, we have assumed that the exponent α
in (A.1) satisfies the condition (n − 2)α ≤ 2, when n ≥ 3. We can still associate
with (4.27) a continuous semigroup S(t) on V , provided (n − 2)α ≤ 4 and that the
domain Ω is either convex or of class C 1,1 . In this case, the proof of the existence of
the associated continuous semigroup S(t) is less straightforward and uses a fixed point
argument introduced by Fujita and Kato (see [FK] and also [GR00]). This semigroup
admits a compact global attractor A in V , with the same properties as above. The
restriction of the above semigroup S(t) to X2 = H 2 (Ω) ∩ H01 (Ω) is also a continuous
semigroup on X2 and A is the global attractor of S(t)|X2 .
2. If we do not want to introduce a limitation on the growth rate of f , we can also
- 83 consider the equation (4.25) in the space X = C00 ≡ {v ∈ C 0 (Ω) | v = 0 in ∂Ω} and
introduce the operator Ac = −∆ with domain D(Ac ) = {v ∈ X ∩ H01 (Ω) | ∆v ∈ X}. If
g = 0 and f (0) = 0, local existence and uniqueness of the solutions of (4.25) are well
known. If, moreover, there exists a positive constant κ such that,
yf (y) ≤ 0 ,
∀|y| ≥ κ ,
(4.34)
one shows, by using the maximum principle and truncature arguments, that the solutions of (4.25) are all global, which allows to associate to (4.25) a continuous semigroup
Sc (t) on X. One proves under the assumption (4.34), that, for any bounded set B ⊂ X,
any τ > 0 and any 0 < β < 1, the orbit γτ+ (B) is bounded in C 1,β (Ω) and hence that
Sc (t) has a compact global attractor in X (see [HK], [Har91] for further details, and
also [Po00] for a setting in the spaces W s,p (Ω), p ≥ 2, 1 ≤ s < 2).
4.4. One-dimensional scalar parabolic equations
In the one-dimensional case, detailed properties of the flow on the compact attractor can
be obtained by using tools like the Sturm-Liouville theory, the Jordan curve theorem as
well as the strong maximum principle. Here, we are going to distinguish two types of
boundary conditions. Because of lack of space, we describe only a few results and refer
to the fairly complete review of Hale [Hal97] for further results.
The case of separated boundary conditions.
For sake of simplicity, we consider the following reaction diffusion equation on Ω = (0, 1),
provided with homogeneous Neumann boundary conditions:
ut = uxx + f (x, u, ux) in Ω = (0, 1) ,
ux (0) = ux (1) = 0 .
(4.35)
We could consider more general separated boundary conditions like
b0 ux (0, t) + β0 u(0, t) = b1 ux (1, t) + β1 u(1, t) = 0 ,
(4.36)
where b0 , β0 , b1 , β1 are normalized so that b20 + β02 = b21 + β12 = 1, and also replace uxx
with a(x)uxx , where a ∈ C 2 (Ω) is a positive function on Ω.
We assume now that f : [0, 1] × R2 → R is a C 2 -function satisfying both conditions:
(C1) there exist γ, 0 ≤ γ < 2, and a continuous function κ : [0, +∞) → [0, +∞)
such that
|f (x, y, ξ)| ≤ κ(r)(1 + |ξ|γ ) ,
∀(x, y, ξ) ∈ [0, 1] × [−r, r] × R ;
(4.37)
- 84 (C2) there exists a positive constant K such that
yf (x, y, 0) ≤ 0 ,
∀(x, y) ∈ [0, 1] × R ,
|y| > K .
(4.38)
Under these hypotheses, the equation (4.35) generates a continuous semigroup S(t) on
the space X s = D((−∆N + I)s ), 1/2 ≤ s ≤ 1, where ∆N is the Laplacian operator
with homogeneous Neumann boundary conditions. We recall that D(−∆N + I) = {u ∈
H 2 (Ω) | ux(0) = ux (1) = 0}. The dissipation condition (C2) implies that the orbits of
bounded sets are bounded.
In the one-dimensional case, the presence of gradient terms in the nonlinearity does
not prevent the gradient structure as it has been proved by Zelenyak [Ze].
Proposition 4.19. The continuous semigroup S(t) is a gradient system on X s , s ∈
[1/2, 1]. Moreover, every positive orbit is convergent.
Sketch of the proof. The first step of the proof consists in finding a Lyapunov
functional Φ0 (u) for (4.35). For u ∈ X s , one considers functionals
Z 1
Φ0 (u) =
G(x, u, ux ) dx,
0
where G : (x, y, ξ) ∈ [0, 1] × R2 7→ G(x, y, ξ) ∈ R2 and one observes that, for any
solution u(x, t) of (4.35),
Z 1
d
Φ0 (u(t)) = −
Gξξ (x, u, ux)u2t dx ,
(4.39)
dt
0
provided the mapping G satisfies
ξGξy − f Gξξ + Gξx = Gy ,
∀(x, y, ξ) ∈ [0, 1] × R2 ,
ut (0, t)Gξ (0, y, 0) = ut (1, t)Gξ (1, y, 0) = 0 ,
∀y ∈ R .
(4.40)
One then shows that there exists a solution G of class C 2 of (4.40) such that Gξξ (x, y, ξ) >
0. Thus, Φ0 is a strict Lyapunov functional and S(t) is a gradient system.
To prove that the ω-limit set ω(ϕ) is a singleton, for every ϕ ∈ X s , we apply the general
result of [HR92b] mentioned in Remarks 4.3. Indeed, for any equilibrium point e ∈ EP of
(4.35), the eigenvalue problem for the linearization of (4.35) at e (called Sturm-Liouville
problem)
λv = vxx + fy (x, e, ex )v + fξ (x, e, ex )vx ,
∀x ∈ (0, 1) ,
vx (0) = vx (1) ,
(4.41)
- 85 has the following well-known properties. All the eigenvalues λj (e) are real and algebraically simple. The (normalized) eigenfunction ϕj (e) corresponding to the j-th
eigenvalue λj (e) has exactly j − 1 zeros. In particular, if 0 is an eigenvalue of (4.41), it
must be simple and the general convergence result of [HR92b] applies.
Probably, the most important property of the scalar equation (4.35), which is the
starting point of the qualitative description of the global attractor A is the transversality
property of the stable and unstable manifolds of the equilibria. This result is due to
Henry [He85b] (see also [An86] for another proof).
Theorem 4.20. If e and e∗ are hyperbolic equilibria of (4.35), then W u (e) is transversal
to W s (e∗ ). Thus, S(t) is a Morse-Smale system and is structurally stable, when the
equilibria are all hyperbolic.
Here we can only give an idea of the proof of this result (for more details, see
[He85b], [An86] and [Hal88]). It involves the following two basic results. For a continuous function v : [0, 1] → R, let z(v) denote the number (possibly infinite) of zeros of
v in (0, 1). We say that a differentiable function v has a multiple zero at x0 ∈ [0, 1]
if v(x0 ) = vx (x0 ) = 0. An application of the maximum principle in two-dimensional
domains together with the Jordan curve theorem yields the following result (see [Ni],
[Mat82], [Che]).
Lemma 4.21. Let v(x, t) ∈ C 0 ([0, +∞), X s) be a solution of the linear nonautonomous
equation
vt = vxx + a(x, t)v + b(x, t)vx ,
x ∈ (0, 1) ,
vx (0) = vx (1) = 0 ,
(4.42)
where a, b are functions in L∞ ((0, 1)×R). Then, if v is not identically zero, the following
properties hold:
(i) z(v(·, t)) is finite for any t > 0;
(ii) z(v(·, t)) is nonincreasing in t;
(iii) if v(x0 , t0 ) = vx (x0 , t0 ) = 0 for some t0 > 0, x0 ∈ (0, 1), then z(v(·, t)) drops strictly
at t = t0 .
Notice that the above lemma holds as well for other separated boundary conditions
and for periodic boundary conditions. The nonincrease of the zero number together with
the above Sturm-Liouville properties imply the following restriction on the connecting
orbits.
- 86 Lemma 4.22. If e ∈ X s , e∗ ∈ X s are hyperbolic equilibrium points of (4.35) and
there is an element u0 ∈ X s such that α(u0 ) = e and ω(u0 ) = e∗ , then dim W u (e) >
dim W u (e∗ ).
The main ingredients of the proof of Lemma 4.22 are as follows. If u(t) is a solution
of (4.35) through u0 , then ut (t) satisfies a linear equation of the form (4.42), whose
coefficients converge exponentially to those of the linearized equations around e and e∗ ,
when t → −∞ and t → +∞ respectively. Since no solution of (4.42) approaches zero
faster than any exponential, one can show that ut (t) → 0 as t → −∞ along the direction
of one of the eigenvectors of the operator ∂ 2 /∂x2 + fy (x, e, ex )I + fξ (x, e, ex )∂/∂x. It
follows from the above Sturm-Liouville properties that z(ut (t)) ≤ ind(e)−1 for t close to
−∞. Likewise, one shows that z(ut (t)) ≥ ind(e∗ ) for t close to +∞. Then Lemma 4.21
implies that dim W u (e) > dim W u (e∗ ).
To complete the proof of Theorem 4.20, one assumes that the manifolds are not
transversal, uses the characterization of the tangent space T W s (e∗ ) of W s (e∗ ) in terms
of the adjoint of the linearized equation around u(t) and argues as in the proof of
Lemma 4.22 for this adjoint equation to show that dim W u (e) < dim W u (e∗ ), which
contradicts Lemma 4.22.
Lemma 4.22 naturally leads to the problem of connecting orbits, when all of the
equilibrium points are hyperbolic. We say that C(e, e∗ ) is an orbit connecting e to e∗
if, for any point u0 ∈ C(e, e∗ ), we have α(u0 ) = e and ω(u0 ) = e∗ . This problem has
been discussed for a long time in the special case of the Chafee-Infante equation:
ut = uxx + µ2 (u − u3 ) in (0, 1) , ux (0) = ux (1) = 0 .
(4.43)
It has been shown by Chafee and Infante [CI] that the only stable equilibrium points of
(4.43) are the constant functions ±1. Furthermore, for each j = 1, 2, · · · two equilibrium
points e±
j of index j bifurcate supercritically from 0 at µj = jπ. In the interval (0, π),
there are three equilibrium points 0, ±1; in the interval (µj , µj+1 ), 0 has index j + 1 and
there are exactly 2j + 3 equilibria 0, ±1, e±
k , k = 1, · · ·, j. The complete description of
the attractor Aµ has been given by Henry in [He85b]. For µ ∈ (µj , µj+1 ), the attractor
Aµ is the closure of W u (0), and, for each 1 ≤ k ≤ j, there exists an orbit connecting
±
e±
k to el , for 1 ≤ l < k, and to ±1.
Before presenting general results on the existence of connecting orbits, we describe
another important consequence of the properties of the zero number; that is the existence of an inertial manifold of (4.35) of minimal dimension, when the equilibria are all
- 87 hyperbolic. More precisely, let N = maxej ∈EP ind(ej ). Using the zero number, Rocha
[Ro91] has shown that, for any equilibria ej and ek , with j 6= k, one has
z(ej − ek ) < N .
As a consequence, he proves the existence of a Lipschitz continuous inertial manifold.
Theorem 4.23. If the hypotheses (C1) and (C2) hold and the equilibria are all hyperbolic, there exists an ( Lipschitzian) inertial manifold of (4.35) of minimal dimension
N and it is a graph over the linearized unstable manifold of maximal dimension. If
f (x, y, ξ) = f (x, y), the inertial manifold is of class C 1 .
This result had been proved before by Jolly [Jo] in the case f (x, y, ξ) = y − y 3 and
by Brunovsky [Br90] in the general case f (x, y, ξ) = f (y).
We now go back to the problem of connecting orbits in the global attractor A
of (4.35). We consider here the semigroup S(t) acting on X 1 and assume that all of
the equilibria are hyperbolic. The case of a nonlinearity f , depending only on u was
solved by Brunovsky and Fiedler ([BrFi89]). Recently, the general case of a nonlinearity
f (x, y, ξ) has been mainly considered by Fusco and Rocha, Fiedler and Rocha, Wolfrum
([Wo]). To determine the set EP of the equilibrium points of (4.35), one solves the ODE
ux = v ,
vx = −f (x, u, v) ,
u(0) = u0 ,
v(0) = 0 .
(4.44)
Since A is compact, the set EP is a finite set of k elements {e1 , . . . , ek } that we have
ordered so that
e1 (0) < e2 (0) < . . . < ek (0) .
(4.45)
By uniqueness of the solutions of (4.44), these values are distinct. At the other boundary
point x = 1, this order may have changed. We thus obtain a permutation πf ≡ π of the
set {1, . . . , k} given by
eπf (1) (1) < eπf (2) (1) < . . . < eπf (k) (1) .
(4.46)
This shooting permutation was introduced by Fusco and Rocha [FuRo]. It characterizes
the existence of connecting orbits as proved by Fiedler and Rocha ([FiRo96]).
Theorem 4.24. Let f (x, y, ξ) be a function in C 2 ([0, 1] × R × R; R) satisfying the
conditions (C1) and (C2). Assume that the corresponding equation (4.35) has only
- 88 hyperbolic equilibria. Let πf be the permutation defined by (4.45) and (4.46).
Then πf determines, in an explicit constructive process, which equilibria are connected
and which are not. In other words, this permutation determines which of the sets
C(ei , ej ) are nonempty.
The proof of Theorem 4.24 uses, in a crucial way, the zero number of differences of
solutions of equations of type (4.35), the transversality of stable and unstable manifolds,
the shooting surface, the above mentioned Sturm-Liouville properties and the Conley
index.
The proof of Theorem 4.24 relies on constructive lemmas that we state, without
comments. The first lemma shows that the existence of a connecting orbit from e to e∗
implies a particular type of cascading.
Lemma 4.25. (Cascading) Under the assumptions of Theorem 4.24, assume that e, e∗
are two equilibria with n = ind(e) − ind(e∗ ) > 0. Then C(e, e∗ ) 6= ∅ if and only if, there
exists a sequence (cascade)
e = v0 ,
v1 , . . . ,
vn = e∗
of equilibria such that, for every j, 0 ≤ j < n, we have:
(i) ind(vj+1 ) = ind(vj ) − 1,
(ii) C(vj , vj+1 ) 6= ∅.
Due to the cascading lemma, it suffices to check all of the possible connections from
e to e∗ , when ind(e) − ind(e∗ ) = 1.
Definition 4.26. If e, e∗ are two equilibria of (4.35) with ind(e) − ind(e∗ ) = 1, we
say that the connections between e and e∗ are blocked if one of the following conditions
holds:
(i) z(e − e∗ ) 6= ind(e∗ ),
(ii) there exists a third equilibrium w with w(0) between e(0) and e∗ (0) such that
z(e − w) = z(e∗ − w) = z(e − e∗ ).
Brunovsky and Fiedler [BrFi89] had already shown that blocking prevents connections. In [FiRo96], the reverse property is proved.
Lemma 4.27. (Liberalism) Let e, e∗ be two equilibria of (4.35) with ind(e) − ind(e∗ ) =
1. Then, C(e, e∗ ) 6= ∅ if and only if the connections from e to e∗ are not blocked.
- 89 Theorem 4.24 raises the question, whether the shooting permutation determines
the global attractor. More precisely, let us denote by Af and πf the global attractor
and the shooting permutation of the semigroup Sf (t) generated by (4.35), corresponding to the nonlinearity f . Fiedler and Rocha [FiRo97] have shown that the shooting
permutation πf completely determines the global attractor, up to an orbit preserving
homeomorphism.
Theorem 4.28. Let f1 (x, y, ξ) and f2 (x, y, ξ) be two functions in C 2 ([0, 1] × R × R; R)
satisfying the conditions (C1) and (C2). Assume that the corresponding equations
(4.35) have only hyperbolic equilibria. Then the equality πf1 = πf2 implies that the
global attractors Af1 and Af2 are topologically equivalent.
Theorem 4.24 and Theorem 4.28 have been given in the frame of homogeneous
Neumann boundary conditions. Although the global attractor for a given nonlinearity
depends on the choice of the boundary conditions, the set of their topological equivalence
classes is independent of the boundary conditions in the following sense. Let us consider
again the reaction diffusion equation
ut = uxx + f (x, u, ux) in (0, 1) .
(4.47)
For τ = (τ0 , τ1 ) given in [0, 1]2 , we provide (4.47) with the boundary conditions:
−τ0 ux (0, t) + (1 − τ0 )u(0, t) = τ1 ux (1, t) + (1 − τ1 )u(1, t) = 0 ,
(4.48)
If f (x, y, ξ) is a function in C 2 ([0, 1] × R × R; R) satisfying the conditions (C1) and
(C2), then the equations (4.47) and (4.48) generate a continuous semigroup Sfτ (t) on
X 1 , which admits a compact global attractor Aτf . Using a homotopy argument together
with the Morse-Smale property of the global attractors, Fiedler [Fi96] has obtained the
equality of the sets of topological equivalence.
Theorem 4.29. Let τ = (τ0 , τ1 ) ∈ [0, 1]2 and σ = (σ0 , σ1 ) ∈ [0, 1]2 be given. If
f (x, y, ξ) is a function in C 2 ([0, 1] × R × R; R) satisfying the conditions (C1) and (C2)
and if Sfτ (t) has only hyperbolic equilibria, then there exists a function g(x, y, ξ) ∈
C 2 ([0, 1] × R × R; R) satisfying the conditions (C1) and (C2), such that Sgσ (t) has
only hyperbolic equilibria and that the global attractors Aτf and Aσg are topologically
equivalent.
- 90 The case of periodic boundary conditions.
If we allow periodic boundary conditions for the reaction-diffusion equation in
(4.35), then the structure of the flow can be different. Let us consider the equation
ut = uxx + f (x, u, ux) , ∀x ∈ S 1 ,
(4.49)
where f : [0, 1] × R2 → R is a C 2 -function satisfying the conditions (C1). The equation
(4.49) defines a local continuous semigroup S(t) on X 1 = H 2 (S 1 ) and, if moreover
the condition (C2) holds, S(t) admits a compact global attractor A. In the case of
separated boundary conditions, we have seen in Proposition 4.19 that the ω-limit set
of any ϕ ∈ X 1 is a singleton. Here we may have closed orbits, as it can be shown in
some explicit examples (see [AnFi88]). Furthermore, Fiedler and Mallet-Paret [FiMP89]
have proved the following somehow surprising generalization of the classical PoincaréBendixson theorem.
Theorem 4.30. If the conditions (C1) and (C2) hold, then the ω-limit set of any
ϕ ∈ X 1 satisfies exactly one of the following alternatives:
(i) either ω(ϕ) consists in precisely one periodic orbit of minimal period p > 0, or
(ii) α(ψ) ⊂ EP and ω(ψ) ⊂ EP , for any ψ ∈ ω(ϕ).
The alternative (ii) means that ω(ϕ) consists of equilibria and connecting (homoclinic or heteroclinic) orbits. Again, the main tool in the proof of Theorem 4.30 is the
zero number (for more details, see [FiMP89] and also [Po00]).
If f is independent of x, all periodic orbits are actually rotating waves, i.e. solutions
of the form u = u(x − ct). Independently, Massat [Ma86] had proved that, in this
particular case, either ω(ϕ) is a single rotating wave or a set of equilibria which differ
only by shifting x. Matano [Mat88] had shown that, if f (y, ξ) = f (y, −ξ), then ω(ϕ) is
a single equilibrium point. In the case where f is analytic in his arguments, Angenent
and Fiedler [AnFi88] had proved before that, if ψ ∈ ω(ϕ), then ω(ψ) and α(ψ) contain
a periodic orbit or an equilibrium point and that every periodic orbit is a rotating wave.
Furthermore, heteroclinic orbits between rotating waves are constructed in [AnFi88].
- 91 4.5. A damped hyperbolic equation
We now illustrate some of the additional difficulties encountered when one considers
partial differential equations which do not smooth in finite time but are still dissipative
and have a global attractor. As a model we choose the linearly damped wave equation,
which arises as mathematical model in biology and in physics ([Za]). The equation
with non linearity f (u) = sin u is called Sine-Gordon equation and is used to model the
dynamics of a Josephson junction driven by a current source. The equation with non
linearity f (u) = |u|α u arises in relativistic quantum mechanics.
The equation with constant positive damping.
We begin the analysis with the following equation with constant positive damping:
∂ 2u
∂u
(x,
t)
+
γ
(x, t) = ∆u(x, t) + f (u(x, t)) + g(x) , x ∈ Ω ,
∂t2
∂t
u(x, t) = 0 , x ∈ ∂Ω , t > 0 ,
∂u
(x, 0) = v0 (x) , x ∈ Ω ,
u(x, 0) = u0 (x) ,
∂t
t>0,
(4.50)
where γ is a positive constant, Ω is a bounded domain in Rn , with Lipschitzian boundary. We assume that g belongs to L2 (Ω) and that f : R → R is a locally Lipschitz
continuous function satisfying the assumption (A.1). As in Section 4.3, we introduce
the operator A = −∆D with domain D(A) = {v ∈ H01 (Ω) | − ∆v ∈ L2 (Ω)} and the
mapping f : v ∈ V 7→ f (v)(x) ∈ L2 (Ω) . We write (4.50) as a system of first order
dU
(t) = BU (t) + f ∗ (U (t)) + G ,
dt
t>0,
U (0) = U0 ,
(4.51)
where
B =
0
I
−A −γ
,
∗
f (U ) =
0
f (u)
,
0
G =
,
g
u
U =
,
v
and introduce the Hilbert space X = V × L2 (Ω) = H01 (Ω) × L2 (Ω), equipped with
the norm kU kX = (k∇uk2L2 + kut k2L2 )1/2 . Since the operator B0 : (u, v) ∈ D(B0 ) 7→
(v, −Au) ∈ X is a skew-adjoint operator on X, where D(B0 ) = D(B) = D(A) × H01 (Ω)
and that f ∗ : X → X is Lipschitz continuous on the bounded sets of X, for any
r > 0, there exists T ≡ T (r) > 0 such that, for any U0 ∈ X, with kU0 kX ≤ r, the
equation (4.51) has a unique mild (or integral) solution U ∈ C 0 ([−T, T ], X). If moreover
- 92 U0 ∈ D(B), then U ∈ C 0 ([−T, T ], D(B))∩C 1([−T, T ], X) is a classical solution of (4.51).
We now introduce the functional Φ ∈ C 0 (X, R) defined by
Φ(U ) = Φ((u, v)) =
Z Ω
1
1 2
v (x) + |∇u(x)|2 − F (u(x)) − g(x)u(x)
2
2
dx .
(4.52)
One easily shows that, if U ∈ C 0 ([0, T ], X) is a solution of (4.51), Φ(U (t)) ∈ C 1 ([0, T ])
and
d
Φ(U (t)) = −γkv(t)k2L2 , ∀t ∈ [0, T ] ,
(4.53)
dt
which implies that Φ is a strict Lyapunov function for (4.51). One remarks that the set
of equilibria EH of (4.51) is given by
EH = EP × {0} = {(u, 0) ∈ X | ∆D u + f (u) + g = 0} .
If f and F satisfy the assumptions (A.1) and (A.3), we deduce from the property
(4.53), by arguing as in (4.33), that all the solutions of (4.51) are global and the orbits
of bounded sets are bounded. If, for any U0 ∈ X, we let S(t)U0 denote the solution
U (t) of (4.51), we have defined a continuous gradient system S(t) on X. In addition,
the mapping (t, U0 ) 7→ S(t)U0 belongs to C 0 ([0, +∞) × X, X).
In 1979, Webb has proved that each positive orbit γ + (U0 ) is relatively compact in
X, by using the variation of constants formula and the arguments leading to the proof
of Remark 2.33 (see [Web79a] and [Web79b]). Actually, under the assumptions (A.1)
and (A.3), the semigroup S(t) has a compact global attractor A. In the non critical
case, that is, under the additional assumption (n − 2)α < 2 when n ≥ 3, the existence
of a compact global attractor has been proved, in 1985, independently by Hale [Hal85]
and Haraux [Har85] (see also [GT87b]). In his proof, Hale showed that the assumptions
of Remark 2.33 are satisfied, whereas Haraux proved that the complete orbits belong
to a more regular space than X, when the domain Ω is more regular. One notices that
the proof of Hale does not require regularity of the domain and also works for more
general operators than the Laplacian, with less regular coefficients. In the critical case
(n − 2)α = 2 when n ≥ 3, the existence of a compact global attractor has been first
given by Babin and Vishik [BV89b] under an additional assumption on f and later by
Arrieta, Carvalho and Hale [ACH] in the general case. Another proof using functionals
has been outlined by Ball [Ba1]. Here, we give a sketch of these proofs and explain their
comparative advantages. We begin with two preliminaries remarks.
- 93 From the assumption (A.1), we at once deduce that
+ kf (0)kL2 ) .
kf (u)kL2 ≤ c0 (kukL2 + kukα+1
L2(α+1)
(4.54)
If Ω is a bounded domain in Rn with Lipschitzian boundary, the embedding from H 1 (Ω)
into L2(α+1) (Ω) is compact, if n = 1, 2 or if n ≥ 3 and (n − 2)α < 2, which implies that
f ∗ : X → X is a compact map.
We also recall that the operator B is the infinitesimal generator of a linear C 0 -group
etB . Using adequate functionals as below or spectral arguments, one shows that there
exist positive constants c1 and c2 such that
ketB kL(X,X) ≤ c1 exp(−c2 γt) ,
t ≥ 0.
(4.55)
The next theorem of existence of a compact global attractor is fundamental.
Theorem 4.31. Assume that the assumptions (A.1) and (A.3) hold. Then, the semigroup S(t) generated by (4.51) has a compact connected global attractor A ⊂ X, given
by A = W u (EP × {0}).
Furthermore, if the domain Ω is either convex or of class C 1,1 , and if the additional
assumption (A.2) holds, then A is compact in X2 = (H 2 (Ω) ∩ V ) × V and is the global
attractor of S(t) restricted to X2 .
Proof. Since S(t) is a gradient system, whose set EH of equilibrium points is bounded
in X, and since the orbits of bounded sets are bounded, we may deduce the existence
of a compact global attractor from Theorem 4.6, as soon as we have shown that S(t) is
asymptotically smooth. We shall present three different proofs of this property.
1. Since S(t)U0 = U (t) is a mild solution of (4.51), one can write
tB
S(t)U0 = e U0 +
Z
t
e(t−s)B (f ∗ (S(s)U0 ) + G) ds .
(4.56)
0
If n = 1, 2 or if n ≥ 3 and (n − 2)α < 2, we deduce from (4.54), (4.55) and (4.56) that
S(t) satisfies the hypotheses of Remark 2.33. Hence S(t) is asymptotically smooth and
has a compact global attractor A in X. As f is actually a bounded map from V into
H s (Ω), for some positive s, one can either use a “bootstrap” argument (see [Har85]) or
apply Theorem 3.18 or Theorem 3.20 to show that, under the additional smoothness
assumptions on f and Ω, the global attractor A is bounded in X2 .
- 94 2. In the case n ≥ 3 and (n − 2)α = 2, the mapping f ∗ : X → X is no longer compact
and a more complicated argument is needed. We first present the functional argument
of [Ba1]. To this end, we introduce the space Y = L2 (Ω) × V ′ . One easily checks that
S(t) is continuous on the bounded subsets of X for the topology of Y and that, for any
bouded set B ⊂ X, the orbit γ + (B) is relatively compact in Y . We set, for any U ∈ X,
Z γ 2
E0 (U ) =
u + uv dx ,
(4.57)
Ω 2
2
2
F (U ) = γE0 (U ) + 2Φ(U ) ≡ kvkL2 + k∇ukL2 + F0 (U ) .
A simple computation and a density argument imply
dF (U (t))
+ γF (U (t)) = F1 (U (t)) ,
dt
(4.58)
where U (t) = S(t)U0 and
F1 (U ) =
Z
(
Ω
γ3 2
u + γ 2 uv + γf (u)u − 2γF (u) − γgu) dx .
2
(4.59)
Integrating (4.58), we obtain the equality (2.21) of Proposition 2.35. Clearly, the functionals F0 and F1 are bounded on the bounded sets of X and continuous on the bounded
sets of X for the topology of Y , not only in the case (n − 2)α ≤ 2, but also in the case
(n − 2)α < 4. Indeed, the continuity of these functionals is proved by showing, that, if
R
un converges to u in L2 (Ω) and is bounded in V , then terms like Ω |un |α+1 |un − u| dx
converge to 0. Using the classical Sobolev inequalities, one gets
Z
s
|un |α+1 |un − u| dx ≤ kun kα+1
ku − un kLq ≤ kun kα+1
(4.60)
H 1 ku − un kH ,
L2n/(n−2)
Ω
R
where 2 ≤ q < 2n/(n − 2) and 0 < s < 1, which implies that Ω |un |α+1 |un − u| dx
converges to 0. By Proposition 2.35, S(t) is asymptotically smooth and hence has a
global attractor A in X. The boundedness of A in X2 is a consequence of [HR00].
3. Unfortunately the previous proof can hardly be generalized to the cases where the
damping term γut is replaced by γ(x)h(ut) with γ(x) ≥ 0 and h(·) a nonlinear adequate
function. In the critical case, the splitting (4.56) of S(t), which is just the linear variation
of constants formula does not directly imply that S(t) is asymptotically smooth. Thus,
as in [BV89b] and in [ACH], we introduce another type of splitting, which relies on a
non linear variation of constants formula (see [ACH] for further details). In [ACH], it
was first remarked that, if f satisfies the conditions (A.1), (A.2) with α1 = 1 and (A.3),
- 95 then f can be written as a sum f = f1 + f2 where f1 et f2 are two functions in C 1 (R, R)
with locally Lipschitzian derivatives and satisfy
|f1′ (y1 ) − f1′ (y2 )| ≤ c1 (1 + |y1 |β1 + |y2 |β1 )|y1 − y2 |α1 , ∀y1 , y2 ∈ R ,
f1′ (0) = 0 ,
|f2 (y)| ≤ c2 ,
yf1 (y) ≤ µ1 y 2 ,
|f2′ (y)| ≤ c2 ,
∀y ∈ R ,
(4.61)
∀y ∈ R ,
µ1 < λ1 ,
where c1 , c2 are positive constants. We now introduce the continuous semigroup S1 (t) :
U0 ∈ X 7→ U1 (t) ∈ X defined by the equation
dU1
(t) = BU1 (t) + f1∗ (U1 (t)) ,
dt
t>0,
U1 (0) = U0 ,
(4.62)
where f1∗ (U1 ) = (0, f1 (u1 )). The mapping S1 (t) is asymptotically contracting, that is,
for any r > 0, there exist positive numbers k1 (r) and k2 (r) such that, if kU0 kX ≤ r, we
have, for any t ≥ 0,
kS1 (t)U0 kX ≤ k1 (r) exp(−k2 (r)t) .
(4.63)
The property (4.63) is easily proved by using the functional E1 (U ) = γE0 (U ) + 2Φ1 (U ),
where Φ1 is nothing else as the functional Φ, in which F (u(x)) + g(x)u(x) has been
R u(x)
f1 (s)ds. Indeed, using the properties (4.61) of f1 , one
replaced by F1 (u(x)) = 0
shows that
ν1 kU1 (t)k2X ≤ E1 (U1 (t)) ≤ (1 + C(r))kU1 (t)k2X ,
d
E1 (t) ≤ −ν1 γkU1 (t)k2X ,
dt
where ν1 = min(1/2, 1 − µ1 /λ1 ). From these inequalities, we easily deduce (4.63).
We next consider the solution U2 (t) = (u2 (t), u2t(t)) = S2 (t)U0 of the following equation
d2 u2
du2
(t)
+
γ
(t) = ∆D u2 (t) + f2 (u(t)) + g + f1 (u(t)) − f1 (u1 (t)) ,
dt2
dt
du2
(0) = 0 ,
u2 (0) = 0 ,
dt
t>0,
(4.64)
Since S(t)U0 = S1 (t)U0 + S2 (t)U0 and that the semigroup S1 (t) is asymptotically contracting, S(t) will be asymptotically smooth, if we show that, for any bounded set B
in X, the set {S2 (t)U0 | U0 ∈ B} is relatively compact in X, for t > 0. Classical energy
estimates arguments show that there exists θ, with 1/2 < θ < 1, such that, for any
U0 ∈ BX (0, r) and any t ≥ 0,
kA1/2−θ/2
du2
(t)kL2 + kA1−θ/2 u2 (t)kL2 ≤ k3 (r) ,
dt
(4.65)
- 96 where k3 (r) is a positive constant depending only on r (for more details, see [BV89b,
Chapter 2, Section 6], [ACH] or [GR00]). This estimate implies in particular that
S2 (t) is a compact mapping for t > 0. It then follows from Theorem 2.31 that S(t)
is asymptotically smooth and that S(t) admits a compact global attractor in X. The
estimate (4.65), which is independent of t > 0, as well that the invariance of A imply
that A is actually bounded in H 2−θ (Ω) × H 1−θ (Ω). Finally a “bootstrap” argument
shows that A is bounded in X2 .
4. Under the additional smoothness hypotheses, the semigroup S(t)X is also bounded
2
dissipative in X2 and asymptotically smooth in X2 (for a detailed proof see [HR88]
or [La86]). The asymptotic smoothness of S(t) in X2 is proved like in 1. Indeed, for
(n − 2)α = 2, f ∗ : X2 → X2 is a compact map. Thus, S(t) has a compact global
attractor A2 in X2 . Obviously, A2 ⊂ A. On the other hand, A2 attracts the bounded
invariant set A and thus, A ⊂ A2 . The theorem is proved.
Remark 4.32. In the part 2 of the proof of Theorem 4.31, we have seen that the critical
exponent for the energy estimates is actually given by (n − 2)α = 4. Unfortunately,
local existence of solutions of the equation (4.51), when Ω is a bounded domain, is not
known if 2 < (n − 2)α < 4. However, due to the Strichartz inequalities, local existence
of solutions of the wave equation in the whole space Rn is known, if 2 < (n − 2)α < 4.
To obtain local existence of solutions of (4.50), one can use these techniques, if one is
able to extend the equation (4.50) to an equation on the whole space in an appropriate
way. This can be done in the case of Neumann boundary conditions for special domains
and in the case of periodic boundary conditions if Ω = (0, L)n, L > 0, for example.
Here let us consider only the case of periodic boundary conditions, when Ω = (0, L)n,
n ≥ 3 and assume that the inequality (4.26), with 2 < (n − 2)α < 4, and the property
(4.29), with µ < 0, hold. Extending the solutions of (4.50) to the whole space Rn and
using the Strichartz inequalities allow to show global existence and uniqueness of the
solutions of (4.50) as well as the continuity of the mapping (t, U0 ) ∈ [0, +∞) × X 7→
S(t)U0 ∈ X. One also shows that S(t) is continuous on the bounded sets of X for the
topology of Y . Like in the case (n − 2)α ≤ 2, S(t) is a gradient system with Lyapunov
functional Φ. Actually, the functional introduced in (4.57) allows to prove that S(t) is
bounded dissipative in X. The same functional argument as in the part 2 of the proof
of Theorem 4.31 implies that S(t) has a compact global attractor A in X (for further
details, see [Ka95], [Fe95], [GR00] and [Ra00]; see also [Lo] for earlier results). In the
case when f satisfies the additional condition (A.2), Kapitanski [Ka95] had proved the
- 97 existence of the compact global attractor A in X by using a splitting method similar
to the one used above and by showing that A is bounded in a more regular space than
X. A “bootstrap” argument finally implies that A is bounded in X2 . In the case of
Ω = Rn , the existence of a compact global attractor in the case 2 < (n − 2)α < 4 had
been proved by Feireisl [Fe95].
Properties of the compact attractor A.
We have seen in Section 3.3 that, under additional conditions, the restriction of the
flow to the compact global attractor A is a more regular function of the time variable.
In the case of the damped wave equation, Theorem 3.18 and Theorem 3.19 apply.
Indeed, H1, H2 and H3 are easily proved and the boundedness of A in H s+1 (Ω) ×
H s (Ω) for some s > 0 implies the compactness condition H5. From Theorem 3.20,
we deduce that the elements in A are more regular functions of the spatial variable x
(see [HR00] for further details). We recall that, in the case of a smooth domain, such
regularity results had been proved by Ghidaglia and Temam [GT87a], when (n−2)α < 2.
Gevrey regularity results for the orbits contained in A in the case of periodic boundary
conditions are also given in [HR00]. For sake of simplicity, we assume in the next
theorem that n = 1, 2, 3 (for details, see [HR00]).
Theorem 4.33. Assume that the conditions (A.1), (A.2) and (A.3) hold and that Ω is
a bounded domain of class C 1,1 in Rn , n = 1, 2, 3.
k
(R, R), k ≥ 1 and f (k) is locally Hölder continuous (resp. f : R → R is a
1. If f ∈ Cbu
real analytic function), then, for any U0 ∈ A, t ∈ R 7→ S(t)U0 ∈ X is of class C k (resp.
analytic).
2. If moreover Ω is of class C k−1,1 and g ∈ H k−1 (Ω), then A is bounded in (H k+1 (Ω) ∩
V ) × H k (Ω).
Since the assumptions H1, H2, H3 and H4 hold, Theorem 3.17 implies that the
equation (4.51) has the property of finite number of determining modes.
Generalizing the results of [CFT85] to the non compact case, Ghidaglia and Temam
have shown that, under the hypotheses (A.1), with (n − 2)α < 2, (A.2) and (A.3),
the global attractor A of (4.51) has finite fractal dimension. If (n − 2)α = 2 or if
2 < (n − 2)α < 4 in the periodic case, the same type of proof shows that A has finite
fractal dimension.
- 98 In what follows, we assume that the three assumptions (A.1), (A.2) and (A.3) hold.
We remark that (e, 0) is a hyperbolic equilibrium point of (4.51) if and only if e is
a hyperbolic equilibrium point of (4.27) and that ind((e, 0)) = ind(e). Indeed, if lj (e),
j ≥ 1, denote the eigenvalues of the operator −A + Df (e), then the eigenvalues of the
operator B + (Df (e))∗ ∈ L(X, X) are given by
q
1
±
−γ ± γ 2 + 4lj (e) , if γ 2 + 4lj (e) ≥ 0 ,
µj =
2
(4.66)
q
1
±
2
2
µj =
−γ ± i |γ + 4lj (e)| , if γ + 4lj (e) < 0 .
2
Thus, if all the equilibrium points (e, 0) are hyperbolic, Theorem 4.6 and Theorem 4.8
imply that
[
A =
W u (ej ) ,
ej ∈EP
and that the Hausdorff dimension dimH (A) is equal to maxe∈EP ind(e). Moreover,
the unstable and stable manifolds W u ((e, 0), S(t)) and W s ((e, 0), S(t)) are embedded
C 1 -submanifolds of X of dimension ind(e) and codimension ind(e), respectively. In
general, one does not know if the stable and unstable manifolds intersect transversally,
even when Ω is an interval of R. As in the parabolic case, we can replace the function
f (·) by a function f (x, ·) depending on the spatial variable x ∈ Ω. If one assumes that
the conditions (A.1), (A.2) and (A.3) hold uniformly in x, then the semigroup S(t) still
admits a compact global attractor A. Generalizing the proof of [BrP97a] to the damped
wave equation, one shows that (4.51) is a Morse-Smale system, generically in the pair
of parameters (γ, f (x, ·)) (see [BrR]).
Unfortunately, even in the one-dimensional case, the orbit structure on A is not
really known. Indeed, unlike the parabolic case, arguments using the zero number
are not applicable. At this time, no good tools seem to be available. Till now, we
do not know, for instance, if A can be written as a graph, nor if it is contained in
a Lipschitzian manifold. Moreover, one does not expect that the stable and unstable
manifolds intersect transversally for all the values of γ. However, if Ω ⊂ R, we deduce
from the Sturm Liouville theorem and from (4.66) that all the eigenvalues of B +
(Df (e))∗ , e ∈ EP , are simple, which, together with Remarks 4.3, implies that all the
orbits of (4.51) are convergent ([HR92b]). Furthermore, if n = 1 and f (u) = µ2 (au −
bu3 ) for example, the bifurcation diagram for the global attractor Aµ with respect to
the parameter µ is essentially the same as the one given by Chafee-Infante for the
- 99 corresponding parabolic equation ([Web79b]). In the case n ≥ 1, all the orbits of (4.51)
are still convergent if f : R → R is an analytic function ([HJ]).
However, one expects that the flow on the global attractor A for γ > 0 very large is
equivalent to the flow on the global attractor of the corresponding parabolic equation,
when this system is Morse-Smale. This is the case, indeed. To prove it, it is easier to
consider the rescaled wave equation
ε
∂ 2 uε
∂uε
(x,
t)
+
(x, t) = ∆uε (x, t) + f (uε (x, t)) + g(x) , x ∈ Ω ,
∂t2
∂t
uε (x, t) = 0 , x ∈ ∂Ω , t > 0 ,
∂uε
(x, 0) = v0 (x) , x ∈ Ω ,
uε (x, 0) = u0 (x) ,
∂t
t>0,
(4.67)
where ε = γ −2 > 0. The formal limit of (4.67) is the parabolic equation (4.25). Hereafter, we denote by Sε (t) the continuous semigroup generated on X by (4.67) and by Aε
the global attractor of Sε (t). We let SP (t) be the semigroup on V , defined by the equation (4.25) and denote by AP the global attractor of SP (t) on V . To compare the attractors Aε and AP , we introduce the set A0 = {(u, v) ∈ X | u ∈ AP , v = ∆u + f (u) + g},
which is bounded in X. If Ω is either convex or of class C 1,1 , A0 is also bounded in X2 .
Beginnning with the papers of Zlamal ([Zl59], [Zl60]) on the telegrapher’s equation
+ uεt − uεxx = 0, the dependence in ε of (4.67) has been extensively analysed (see
[BV87], [HR88], [HR90], [MSM], [Ko90],[Wit] and [Ra99]).
At first glance, (4.67) appears as a singular perturbation of the equation (4.25). Actually, it is not the case, if we compare adequate time-τ maps instead of comparing the
continuous semigroups. For any (u0 , v0 ) ∈ X, we write the solution uε (t) of (4.67) as
uε = uε1 + uε2 , where uε1 and uε2 are the solutions of
εuεtt
εuε1tt + uε1t + Auε1 = 0 ,
(uε1 (0), uε1t )(0) = (0, v0 ) ,
εuε2tt + uε2t + Auε2 = f (uε ) + g ,
(uε2 (0), uε2t)(0) = (u0 , 0) .
(4.68)
Using a priori estimates on (uε1 , uε1t )(t) and comparing (uε2 , uε2t )(t) with (u(t), ut (t)),
where u(t) = SP (t)u0 , we obtain the following result [MR], [Ra99]:
Lemma 4.34. There exist a positive constant ε0 and, for any r > 0, a positive number
C(r), such that, for 0 ≤ ε ≤ ε0 and, for any (u0 , v0 ) ∈ X, satisfying ku0 k2H 1 (Ω) +
εkv0 k2L2 (Ω) ≤ r 2 , we have, for t ≥ 0,
εk
d
(tuε (t) − tu(t))k2L2 + kt(uε − u)(t))k2H 1 ≤ C(r)ε2 (1 + k(u0 , v0 )k2X ) exp C(r)t ,
dt
- 100 where uε and u are the solutions of (4.67) and (4.25) respectively.
Similar estimates hold for the linearized semigroups DSε (t) and DS0 (t). Lemma 4.34
leads to define the semigroup S0 (t) on X by
S0 (t)(u0 , v0 ) = (SP (t)u0 ,
d
(SP (t)u0 )) ,
dt
t>0,
S0 (0)(u0 , v0 ) = (u0 , v0 ) .
(4.69)
Due to the smoothing properties of the parabolic equation (4.25), S0 ∈ C 0 ((0, +∞) ×
X, X) and, for t ≥ 0, S0 (t) ∈ C 1 (X, X). Clearly, S0 (t) is a gradient system with
R
Lyapunov functional Φ0 (u, v) = Ω (1/2|A1/2 u|2 − F (u) − g(x)u)dx and A0 is the global
attractor of S0 (t). For τ > 0 a fixed number, we introduce the C 1 -mapping Sε = Sε (τ )
from X into X, for ε ≥ 0. Lemma 4.34 and its analogue for the linearized semigroups
DSε (t) and DS0 (t) imply that Sε converges to S0 in a C 1 -sense, when ε goes to 0. In
particular, there exists a positive constant C0 (r, τ ) depending only on r and τ so that,
if k(u0 , v0 )kX ≤ r,
kSε (u0 , v0 )−S0 (u0 , v0 )kX +kDSε (u0 , v0 )−DS0 (u0 , v0 )kL(X,X) ≤ C0 (r, τ )ε1/2 . (4.70)
As a direct consequence of (4.70), Theorem 4.11 and Theorem 4.12, we obtain the next
result.
Theorem 4.35. (i) The global attractors Aε are upper semicontinuous at ε = 0.
(ii) If all the equilibrium points e of (4.25) are hyperbolic, the global attractors Aε are
lower semicontinuous at ε = 0 and there exist positive constants ε1 , C and κ ≤ 1/2,
such that, for 0 ≤ ε ≤ ε1 ,
δX (A0 , Aε ) + δX (Aε , A0 ) ≤ Cεκ .
(iii) Assume that the continuous semigroup SP (t) is a Morse-Smale system. Then,
there exist positive numbers τ and ε2 , such that, for 0 ≤ ε ≤ ε2 , Sε (t) is a Morse-Smale
system and there is a homeomorphism hε : A0 → Aε satisfying the conjugacy condition
hε ◦ S0 (τ ) = Sε (τ ) ◦ hε , for any ε > 0.
This example illustrates well the relevance of replacing the comparison of continuous
semigroups by the one of maps.
For more details, we refer to [Ra99], [Ra00] and [MR]. Assertions (i) and (ii) had
been proved earlier in [HR88] and [HR90] respectively (see also [Ko90]). Using the
- 101 assertion (iii) in the case n = 1, we obtain, for ε small enough, the same orbit structure
on Aε as in the parabolic case. In the case n = 1, Mora and Solà Morales [MSM] had
proved that, for ε small enough, the semigroup Sε (t) admits an inertial manifold and
that this inertial manifold converges, in a C 1 -sense, to the one of S0 (t), when ε goes to 0,
which reduces the comparison of (4.67) and (4.25) to a finite-dimensional perturbation
problem. This result can also be deduced directly from the general theorems of C 1 dependence of inertial manifolds with respect to parameters.
Remarks 4.36.
1. All the above assertions (except Remark 4.32) remain true if we replace the homogeneous Dirichlet boundary conditions in (4.50) by homogeneous Neumann boundary
conditions. In this case, V = H 1 (Ω). With some small changes, these assertions also
hold even if we consider more general boundary conditions.
2. If we replace the equation (4.50) by a system of m damped wave equations, that is
not necessarily gradient, one can still show the existence of a compact global attractor
in (V × L2 (Ω))m , under adequate dissipative hypotheses on the non linearity f . In
this case, one shows directly that the associated semigroup is bounded dissipative (see
[Hal88] and [Te]).
du
3. If, in (4.50), one replaces γ du
dt by γ(−∆ + Id) dt , one obtains the so called strongly
damped wave equation. The linear operator B : (u, v) ∈ D(A) × D(A) 7→ (v, −Au −
Av) ∈ X generates an analytic semigroup on X. Under the conditions (A.1) and (A.3),
one shows that the corresponding nonlinear semigroup S(t) can be written in the form
(2.16) and has a compact global attractor in X (see [Web80], [Fit81], [Ma83b], [Hal88]).
In the one-dimensional case, all the orbits are convergent [HR92b].
The equation with a variable non negative damping.
One can now wonder what happens if the damping term γut in (4.50) is replaced by
a function h(ut ) or more generally by γ(x)h(ut ), where γ(x) is a non negative function
on the spatial variable x. The existence of a compact global attractor A in X has been
proved by Ceron and Lopes [CeLo], under the assumptions (A.1), with (n − 2)α < 2,
and (A.3), in the case where γ > 0 is a constant and h ∈ C 1 (R, R) satisfies
h(0) = 0 ,
0 < a ≤ h′ (y) ≤ b ,
∀y ∈ R ,
(4.71)
where a, b are positive constants. In their proof, they introduced the criterium (2.20)
of asymptotic smoothness and applied Proposition 2.34 (for a generalization to the case
- 102 γ(x)h(ut ) and (n − 2)α ≤ 2, see [FZ]).
We now try to present the difficulties encountered when the positive constant γ
is replaced by a nonnegative function γ(x) ∈ C 1 (Ω, [0, +∞)), which is not identically
zero on the closure of Ω. Assume that the conditions (A.1) and (A.3) are satisfied. In
this case, global existence and uniqueness of mild solutions of (4.51), with γ replaced
by γ(x), still hold. As before, we denote by S(t) the associated semigroup on X. If Φ
is the functional introduced in (4.52) and U ∈ C 0 ([0, T ], X) is a solution of (4.51), the
equality (4.53) becomes
d
Φ(U (t)) = −
dt
Z
γ(x)|ut(t, x)|2 dx ,
Ω
∀t ∈ [0, T ] ,
(4.72)
which implies, as in the case of a constant positive damping, that the orbits of bounded
sets are bounded. Unfortunately, without additional conditions on γ(x), we cannot
deduce from (4.72) that Φ is a strict Lyapunov functional.
As a direct consequence of Proposition 2.39, Theorem 2.26 and the part 1 of the proof
of Theorem 4.31, we obtain the following result.
Proposition 4.37. Assume that the hypotheses (A.1), (A.3) and (n − 2)α < 2 hold.
If we suppose that there exist positive constants K and θ such that
keBt kL(X,X) ≤ Ke−θt ,
∀t ≥ 0 ,
(4.73)
then the semigroup S(t) is asymptotically smooth and has a minimal global B-attractor
AX . If, in addition, S(t) is point dissipative, then AX is the compact global attractor
of S(t).
To apply Proposition 4.37, one must first obtain conditions that will imply that
the linear semigroup eBt satisfies (4.73). If γ(x0 ) > 0 at some point x0 ∈ Ω, then each
solution eBt U0 approaches zero as t → +∞ (see [Iw], [Da78]). However, as remarked by
Dafermos [Da78], one can construct examples, for n ≥ 2, where the approach to zero is
not uniform with respect to initial data in a ball and so (4.73) is not satisfied. Using
geometric optics arguments, Bardos, Lebeau and Rauch [BLR] have shown that, if Ω
and γ are of class C ∞ , the property (4.73) holds if the following condition is satisfied:
(BLR) There exists τ > 0 such that every ray of geometric optics intersects the set
s(γ) × (0, τ ), where s(γ) is the support of γ.
- 103 The condition (BLR) is true in particular if s(γ) is a neighbourhood of ∂Ω. Condition (BLR) gives a very interesting way to verify (4.73). However, the question of
characterizing, for a particular domain, the minimal conditions on the damping γ for
which (BLR) holds, is not easy. If
I ⊂ Ω are two intervals of R ,
γ(x) ≥ 0 , ∀x ∈ Ω ,
γ(x) > 0 , ∀x ∈ I ,
(4.74)
then (4.73) holds. Other approaches to prove the property (4.73) are given in [CFNS]
and [Har89], for example (see also [FZ], [Zu90] and the references therein).
It remains to derive conditions on γ(x), which will imply that S(t) is point dissipative. From (4.72), we deduce that, for any U0 ∈ X, ω(U0 ) must be a subset of the
bounded complete orbits of the system
utt (x, t) − ∆u(x, t) = f (u(x, t)) + g(x) ,
ut (x, t) = 0 ,
x ∈ Ω\s(γ) , t ∈ R ,
x ∈ s(γ) , t ∈ R ,
(4.75)
u(x, t) = 0 , x ∈ ∂Ω , t ∈ R .
We now distinguish the cases n = 1 and n ≥ 2.
If n = 1, using the classical representation formula of the solution of a wave equation, we show that, if the condition (4.74) holds, any bounded complete orbit of (4.75)
is an equilibrium point of (4.51). Thus, in this case, Proposition 4.37 implies that (4.51)
has a compact global attractor ([HR93a]). Moreover, one shows that the orbits of (4.51)
are convergent (see [HR93a] and [Ra95] for further examples of convergence in locally
damped wave equations).
If n ≥ 2, we remark that, for any bounded complete orbit (u(t), ut(t)) of (4.75),
Df (u(t)) belongs to Cb0 (R, Ln (Ω)) and w = ut ∈ Cb1 (R, V ′ ) ∩ C 0 (R, L2 (Ω)) is a solution
of the system
wtt (x, t) − ∆w(x, t) − Df (u(x, t))w(x, t) = 0 ,
w(x, t) = wt (x, t) = 0 ,
x∈Ω, t∈R,
x ∈ s(γ) , t ∈ R .
(4.76)
If the only solution w(t) ∈ Cb1 (R, V ′ ) ∩ C 0 (R, L2 (Ω)) of (4.76) is w = 0, then the ω-limit
set of any solution of (4.51) is an equilibrium point. We are thus led to the following
unique continuation property (u.c.p.):
- 104 (u.c.p.) Assume that w is a weak L2 (Ω × (0, T )) solution of the equation
wtt − ∆w + b(x, t)w = 0 in Ω × (0, T )
where T > diam Ω and b ∈ L∞ ((0, T ), Ln(Ω)).
Then, if w vanishes in some set O × (0, T ), O ⊂ Ω, w must be identically zero.
Ruiz [Ru] has shown that the (u.c.p.) property holds when O is a neighbourhood of ∂Ω.
It follows from the above discussion and from Proposition 4.37 that, if ∂Ω ⊂ s(γ)
and (n − 2)α < 2, then the semigroup S(t) has a compact global attractor in X. Feireisl
and Zuazua [FZ] have generalized this existence result to the critical case (n − 2)α = 2.
In their proof, they have used energy functionals arguments to show that S(t) is bounded
dissipative and the same splitting as in Part 3 of the proof of Theorem 4.31 to show
that S(t) is asymptotically smooth. We thus can state the following result (see [HR00]
for the regularity results):
Theorem 4.38. Assume that Ω is a bounded regular domain in Rn . If the hypotheses
(A.1), (A.2), (A.3) are satisfied and if, either the conditions n = 1 and (4.74) hold, or
s(γ) is a neighbourhood of ∂Ω, then (4.51) has a connected compact global attractor
A = W u (EP × {0}) in X. Moreover, the time and spatial regularity properties of the
complete orbits in A, given in Theorem 4.33, still hold and the property of finite number
of determining modes remains true.
Finally, we note that many other wellknown dissipative gradient systems, having
a compact global attractor, could have been approached. Among them, we quote the
strongly damped wave equation (see Remarks 4.36), the Cahn-Hilliard equation (see
[Te] and the references therein), nonlinear diffusion systems ([Hal88]) etc . . ..
5. Further topics
So far, we have mainly studied equations, which have a gradient structure. The
most famous and most studied non gradient dissipative system arising in PDE’s is
certainly the one generated by the Navier-Stokes equations on a bounded domain in
space dimension two or three. As already shown by Ladyzenskaya in 1972 ([La72],
[La73]), in space dimension two, this equation has a compact global attractor, which
is of finite fractal dimension ([MP76], [FT79], [La82]). The associated semi-flow is a
smooth function of the time variable for t > 0 (up to analyticity) and the global attractor
- 105 is composed of smooth functions in the spatial variable (see [FT79], [FT89], [FeTi98],
[Te] and the references therein). Estimates of the fractal and Hausdorff dimensions of
the attractor in terms of various physical parameters have been extensively studied (see
[BV83], [CF85], [CFT85], [La87a], [EFT], [JTi93]). In the two-dimensional case, the
Navier-Stokes equations have also the property of finite number of determining modes
(see [FP67], [La72], [La87b], [CJTi97]). For further details and study on the NavierStokes equations, we refer to [BN] in this volume.
Among the wellknown evolutionary partial differential equations, which have smoothing
properties in finite positive time and admit a compact global attractor, we should also
mention the one-dimensional Kuramoto-Sivashinsky equation (see [NST], [CEES] for
example), modeling pattern formation in thermohydraulics and also the propagation of
a front flame, as well as the complex Ginzburg-Landau equation in space dimensions
one or two, describing the finite amplitude evolution of instability waves. The complex
Ginzburg-Landau equation is actually a strongly damped Schrödinger equation; in space
dimensions one or two, it admits a compact global attractor of finite fractal dimension
([GhHe], [Te]).
To conclude this paper, we present the weakly damped Schrödinger equation, which
is a system generated by a dispersive equation with weak damping.
A weakly damped Schrödinger equation.
In what follows, Ω denotes, either the whole space Rn , n = 1, 2 or 3 or a bounded
C 2 -polygonal domain in Rn , when n = 1, 2. For γ > 0 a fixed constant, f a function
in L2 (Ω) and g ∈ C 1 ([0, +∞), R) a function satisfying the hypotheses (H.1) and (H.2)
below, we consider the weakly damped Schrödinger equation, which arises in plasma
physics or in optical fibers models (see [NB], for instance):
iut + ∆u + g(|u|2 )u + iγu = f ,
in Ω × (0, +∞) ,
u(0) = u0 ,
in Ω
.
(5.1)
If Ω 6= Rn , we associate homogeneous Dirichlet boundary conditions to (5.1)
u = 0,
on ∂Ω .
(5.2)
Of course, we could consider homogeneous Neumann boundary conditions or periodic
Ry
conditions as well. We assume that g ∈ C 1 ([0, +∞), R) and G(y) = 0 g(s) ds satisfy
the following conditions:
- 106 (H.1) there exist two constants C1 > 0 and α1 ∈ [0, 2/n) such that
G(y) ≤ C1 y(1 + y α1 ) ,
yg(y) − G(y) ≤ C1 y(1 + y α1 ) ,
y≥0,
y≥0,
(5.3)
(H.2) in the case when n = 2 or 3, there exist two constants C2 > 0 and α2 ≥ 0,
with (n − 2)α2 < 4, such that, for any (ξ, ξ ′) ∈ C2 ,
|g(|ξ|2)ξ − g(|ξ ′|2 )ξ ′ | ≤ C2 (1 + |ξ|α2 + |ξ ′ |α2 )|ξ − ξ ′ | .
(5.4)
(H.3) in the case when Ω is a bounded subset of R2 , there exists a positive constant
C3 such that
|g ′ (y)| ≤ C3 , y ≥ 0 .
(5.5)
Later we shall also impose the next additional conditions on g, which mainly require
that the nonlinearity is subcritical:
(H.4) The function g is in C ∞ ([0, +∞), R) and there exist two constants C4 > 0 and
α1 ∈ (0, 2/n) such that
y|g ′ (y)| + |g(y)| ≤ C4 y α1 , ∀y ≥ 0 ,
(5.6)
Moreover, for k ≥ 2, the derivatives g (k) are bounded.
As before, we denote by A = −∆BC u the unbounded operator on H = L2 (Ω),
where ∆BC is the Laplace operator with the corresponding boundary conditions. We
set V 2 = D(A), V = D(A1/2 ) and we denote by V ′ the dual space of V .
Proposition 5.1. Under the assumptions (H.1), (H.2) and (H.3), for any u0 ∈ V ,
there exists a unique solution u(t) ∈ C 0 ([0, +∞), V ) of (5.1) and (5.2). Moreover,
u(t) ∈ C 1 ([0, +∞), V ′ ) and the mapping S(t)u0 = u(t) defines a continuous semigroup
on V . If u0 ∈ V 2 , then u(t) belongs to C 0 ([0, +∞), V 2 ) ∩ C 1 ([0, +∞), H).
Furthermore, for any t ≥ 0, the mapping S(t) is continuous on the bounded sets of V
for the topology of H.
Proof. In the case where Ω = Rn , mutatis mutandis, we can follow the proofs of
[CH, Theorem 7.4.1 and Proposition 7.5.1]. These proofs use the well known Strichartz
inequalities. The above results are shown in [Gh88a], when Ω is a bounded domain in R.
- 107 For Ω a bounded domain in R2 , the proofs can be found in [Ab1] (see also [Ab2]). The
proof of the uniqueness of the solution in V is more delicate than in dimension 1 and
requires the hypothesis (H.3). Indeed, in the case of a bounded domain, estimates similar
to Strichartz inequalities are not yet known. The continuity of S(t) on the bounded sets
of V for the topology of H can be shown by arguing as in [CH, Proposition 7.4.2].
Uniqueness of solutions of (5.1) is not known, when Ω is a bounded domain in R3 .
In [Gh88a], it was first proved that, if Ω is a bounded domain in R, then S(t) has a
global weak attractor A1 (resp. A2 ) in V (resp. D(A)) (see Remark 2.30). Using the
functionals given below and applying Proposition 2.35, Abounouh (resp. Laurençot)
have showed the existence of a compact global attractor A in V , when Ω is a bounded
domain of R2 (resp. Ω = Rn ).
We introduce the functionals Φ, Φ0 and Ψ defined on V by
Z
2
Φ(v) = k∇vkH + (−G(|v|2 ) + 2Re (f v))dx ≡ k∇vk2H + Φ0 (v) ,
Ω
Z
Ψ(v) =
(g(|v|2)|v|2 − G(|v|2 ) + Re (f v))dx .
(5.7)
Ω
Obviously, due to the hypothesis (H.2), the functionals Φ0 and Ψ are continuous on the
bounded sets of V for the topology of H. Taking successively the inner product of (5.1)
with u and ut , one shows (see [Ab2], [Gh88a] and [Lau]) that, for any u0 ∈ H 2 (Ω) ∩ V ,
S(t)u0 = u(t) satisfies, for t ≥ 0,
Z
d
2
2
ku(t)kH + 2γku(t)kH = 2
Im (f u(x, t))dx ,
dt
Ω
.
(5.8)
d
Φ(u(t)) + 2γΦ(u(t)) = 2γΨ(u(t))
dt
which implies that, for t ≥ 0,
Φ(S(t)u0 ) = exp(−2γt)Φ(u0 ) + 2γ
Z
0
t
e2γ(s−t) Ψ(S(s)u0 )ds ,
∀u0 ∈ V .
(5.9)
From the hypotheses (H.1), (H.2) and from the equalities (5.8), one deduces that r > 0
can be chosen so that the ball BV (0, r) is positively invariant under S(t) and is an
absorbing set for the semigroup S(t). The above properties lead to the following result.
Theorem 5.2.
1. Under the hypotheses (H.1), (H.2) and (H.3), the semigroup S(t) has a connected,
- 108 compact global attractor A in V .
2. Suppose that the condition (H.4) holds. If Ω is either the whole space Rn , n = 1, 2 or
a bounded interval of R, the global attractor A of (5.1) and (5.2) is compact in H 2 (Ω).
Moreover, in the one-dimensional case, for any u0 ∈ A, the mapping t ∈ R 7→ S(t)u0 is
of class C k , for any k ≥ 0 (resp. analytic if g is analytic). If in addition, f ∈ H k (Ω),
then A is bounded in H k+2 (Ω).
Proof. 1. The first statement is a direct consequence of Theorem 2.26, Proposition 2.20,
if we show that S(t) is asymptotically smooth in V . Since the mapping S(t), for t ≥
0, and the functionals Φ0 and Ψ are continuous on the bounded sets of V for the
topology of H, we apply Proposition 2.35 with X = V , Y = H, F0 = Φ0 , F1 (v) =
R
2γΨ(v) + 2 Ω Im (f v)dx. If Ω is a bounded domain in Rn , n = 1, 2, the condition (ii)
of Proposition 2.35 is clearly satisfied, since the Sobolev imbedding H 1 (Ω) ֒→ L2 (Ω) is
compact. When Ω = Rn , n = 1, 2 or 3, the condition (ii) is proved in [Lau, Lemmas
2.6 , 2.7, 2.8], by using a splitting of the solutions like in [Fe94] or [Fe95].
2. The first part of the proof does not indicate if the compact global attractor is
bounded or compact in a more regular space. When the hypothesis (H.4) holds, the
boundedness of A in H 2 (Ω) is shown by Goubet in [Go96] and [Go98], by using a
splitting of S(t)u into a low wavenumber part PN (S(t)u) and a high wavenumber part
QN (S(t)u). The low wavenumber part is obviously smooth and the high wavenumber
part can be approximated asymptotically by the solution of an equation with zero
initial data. The compactness in H 2 (Ω) then follows from the fact that A is a bounded
invariant set in H 2 (Ω) and thus in contained in the compact global attractor of S(t)
in H 2 (Ω). When Ω is bounded interval of R, the boundedness of A in H k+2 (Ω) is
proved in the same way in [Go96]. The C k -regularity (resp. the analyticity) of the map
t ∈ R 7→ S(t)u0 , for any k ≥ 0 and any u0 ∈ A, is shown in [HR00] as a consequence of
a generalized version of Theorem 3.18 and Theorem 3.19. We note that, in this proof,
the hypothesis (H.4) can be slightly relaxed.
Remarks.
1. In the one-dimensional case, under a relaxed version of the hypothesis (H.4), the
system generated by (5.1) and (5.2) has the property of finite number of determining
modes (see [OTi], [GR00] and [HR00]).
2. In the one-dimensional case, the global attractor of the equation (5.1) with periodic
boundary conditions is regular in the same Gevrey class as g and f and thus is analytic
- 109 in the spatial variable (see [OTi], [HR00]).
3. Goubet [Go99a] has also proved the compactness of the global attractor of the
equation (5.1) with periodic boundary conditions in the two-dimensional case. There
the proof is more involved and uses spaces introduced by Bourgain.
4. From [Gh88a, Theorem 3.2 and Remark 3.1], it follows that in the one-dimensional
case, A has finite fractal dimension. Adapting these proofs, one can certainly show that,
in the other cases considered in Theorem 5.2, A has also finite fractal dimension.
Finally, we notice that the existence and regularity of the compact global attractor
for other weakly damped dispersive equations like the weakly damped KdV and Zakharov equations are proved by using similar methods (see [Gh88b], [GoMo], [MRW]
and [Go99b]).
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